ITERATED INTEGRALS AND CYCLES ON ALGEBRAIC MANIFOLDS
NANKAI TRACTS IN MATHEMATICS Series Editors: Shiing-shen Chern, Yiming Long, and Weiping Zhang Nankai Institute of Mathematics
Published VOl. 1
Scissors Congruences, Group Homology and Characteristic Classes by J. L. Dupont
VOl. 2
The Index Theorem and the Heat Equation Method by Y. L.Yu
VOl. 3
Least Action Principle of Crystal Formation of Dense Packing Type and Kepler’s Conjecture by W. Y. Hsiang
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Lectures on Chern-Weil Theory and Witten Deformations by W. P. Zhang
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Contemporary Trends in Algebraic Geometry and Algebraic Topology edited by Shiing-Shen Chern, Lei Fu & Richard Hain
VOl. 7
Iterated Integrals and Cycles on Algebraic Manifolds by Bruno Harris
VOl. 8
Minimal Submanifolds and Related Topics by Yuanlong Xin
Nankai Tracts in Mathematics - Vol. 7
ITERATED INTEGRALS AND CYCLES ON ALGEBRAIC MANIFOLDS
Bruno Harris Department of Mathematics Brown University, USA
orld Scientific NewJersey London Singapore Hong Kong
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ITERATED INTEGRALS AND CYCLES ON ALGEBRAIC MANIFOLDS Nankai Tracts in Mathematics -Vol. 7 Copyright 0 2004 by World Scientific Publishing Co. Re. Ltd. All rights reserved. This book, or parts thereoj may not be reproduced in anyform or by any means, electronic or mechanical, includingphotocopying, recording or any informationstorage and retrieval system now known or to be invented, without wrinenpermissionfram the Publisher.
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To Professor S.S. Chern and
to the memory of Professor K.T. Chen
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P1:e face
This book begins with some work of K.T.Chen, who used iterated integrals of 1-forms on a manifold to study its fundamental group T I , more precisely the quotients of T I by the sequence of groups in its lower central series: 7r1/(7rll T I ) , ~ 1 / ( ( 7 r 1T, I ) ,T I ) , and so on, (,) denoting the commutator subgroup. The first step is well-known: if we choose a vector space (over R) consisting of closed 1-forms and isomorphic to the first deRham cohomology group of the manifold X then integration of these forms over closed paths gives a homomorphism of r l ( X ) to the R-dual of H 1 ( X ) ,which is just H1 ( X ,R) if we assume H1 finite dimensional, and with this assumption 7r1/(7rl, T I ) (mod torsion) embeds as a discrete cocompact subgroup in the Lie group H l ( X , R ) . Chen showed that if one uses iterated integrals of the closed 1-forms, e.g integrals of the form
in the case of two 1-forms, then one can similarly map 7r1/((7r1 , T I ) , .. . , " 1 ) to a nilpotent Lie group (constructed using H 1 ( X ) and H 2 ( X ) ) and obtain information on this quotient of 7r1; again the image is discrete and cocompact . We will explain Chen's results in the most concrete form for a compact Kahler manifold X , where the whole sequence of T I quotients and their homomorphism to Lie groups is obtained by means of a flat connection 8 canonically associated to X . 0 is a 1-form on X with values in an (infinitedimensional) Lie algebra (or more precisely a formal series of 1-forms on X with Lie algebra coefficients). This is done in Chapter 1, and Chapter 2 studies more closely the special case where X is a compact Riemann surface and only the part of the above construction involving iterated integrals of two 1-forms is examined. It turns out that these iterated integrals give vii
viii
Iterated Integrals and Cycles o n Algebraic Manifolds
information about how the Riemann surface X is embedded in its Jacobian J ( X ) . Following A.Weil [Weil 1962, p.3311 we consider both X and its image X - under the map of J ( X ) to itself which is the group-theoretic inverse , and form the algebraic l-cycle X - X - , which is homologous to 0 in J ( X ) . This l-cycle, following Hodge and Weil, has an image in another torus, an intermediate Jacobian of J ( X ) associated to H 3 ( J ( X ) ) .Weil puts this example in a discussion of whether this algebraic l-cycle homologous t o zero is also algebraically equivalent to 0 (roughly speaking, whether it can be deformed to 0 using an algebraic deformation). We show that just this “quadratic part” of Chen’s construction calculates (gives a formula for) the “Abel-Jacobi” image of X - X - in the intermediate Jacobian. This allows us to prove that “in general” X - X - is not algebraically equivalent to 0 and also to give specific examples, the first specific examples of cycles homologous but not algebraically equivalent t o 0 , consisting of algebraic curves over Zsuch as the Fermat curve x4 y4 = 1. We refer to our papers [Harris 1983a], [Harris 1983133. Chapter 3 (partially) generalizes Chapter 2 to higher dimensional Kahler manifolds X . In the Riemann surface case the construction in Chapter 2 associates to three elements of HI ( X ,Z) represented by mutually disjoint 1-cycles C1, C2,C3 a real number obtained by iterated integrals over one of them, say C3, of harmonic forms ~ 1 a2 , Poincare dual t o C1, C2. If we take twice this real number and reduce it mod Z, and do this for all possible triples of homology classes as above, we can regard this set of real numbers mod Zas the Abel-Jacobi image of X - X - . The generalization in Chapter 3 consists in taking k homology classes [Cl],. , . , [ C k ] (of possibly different dimensions) representable by cycles C1, . . . , ck such that any k - 1 of the Ci are disjoint (so k 2 3) and
+
and associating to this k-tuple of homology classes a real number, which depends only on the complex structure of X (if X has a Kahler metric). This real number is constructed by using the heat kernel exp(-tA) of the Laplace operator A on differential forms. We show that this same number can be expresses by iterated integrals involving the harmonic forms Poincare dual to the [Ci],where the domains of integration are now intersections of the cycles Ci. We have attempted throughout the book to give definitions and details of proofs that will make it accessible to, say, second year graduate students, or perhaps even students with less background. We do however
Preface
ix
assume some acquaintance with topology and with Lie groups in Chapter 1. In Chapter 3 we explain a connection between the heat kernel and cycle pairings on a Riemannian manifold and rely considerably on the second chapter of “Heat Kernels and Dirac Operators” by Berline, Getzler, and Vergne. For K.T.Chen’s work we can refer to his collected works and the article by Richard Hain (see Bibliography). I would like to express my gratitude to Professor Chern Shiing-Shen for inviting me to present these results at the Conference on the Work of K.T.Chen and W.L.Chow at the Nankai Mathematical Institute in October 2000, to come back in Fall 2001 to give a course on this subject, and to write the present book. Professor Chern’s warm kindness and that of other faculty and students at Nankai during this visit have made it an unforgettable experience: I would like to mention especially Professors Zhou Xing-Wei, Bai Chengming, Feng Huitao, Long Yiming, Fang Fuquan, Ge Molin, and students Miss Long Jing and Miss Zhu Tong. For a similar course a t Brown in Fall 2002, I would like to thank Amir Jafari, Wang Qingxue, Justin Corvino, and Alan Landman. For some of the basic mathematical knowledge in this book, I am grateful to my friends H.C.Wang, Gerard Washnitzer, and Ezra Getzler.
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Contents
Preface
vii
1. Iterated Integrals. Chen’s Flat Connection and
~1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Differential equations . . . . . . . . . . . . . . . . . . . . . . 1.3 Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Chen’s Lie algebra and connection . . . . . . . . . . . . . . 1.6 Some work of Quillen . . . . . . . . . . . . . . . . . . . . . . 1.7 Group homology . . . . . . . . . . . . . . . . . . . . . . . . 1.8 The basic isomorphisms . . . . . . . . . . . . . . . . . . . . 1.9 Lattices in nilpotent Lie groups . . . . . . . . . . . . . . . . 1.10 Some Hodge theory . . . . . . . . . . . . . . . . . . . . . . 2 . Iterated Integrals on Compact Riemann Surfaces
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . Generalities on Riemann surfaces and iterated integrals . . Harmonic volumes and iterated integrals . . . . . . . . . . . Use of the Jacobian . . . . . . . . . . . . . . . . . . . . . . . Variational formula for harmonic volume . . . . . . . . . . . 2.6 Algebraic equivalence and homological equivalence of algebraic cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Calculations for the degree 4 Fermat Curve . . . . . . . . . 2.8 Currents and Hodge theory . . . . . . . . . . . . . . . . . . 2.1 2.2 2.3 2.4 2.5
3. The Generalized Linking Pairing and the Heat Kernel xi
1
1 2 7
8 14 19 21 25 26 28 35 35 35 42 45 47 52 55 67 73
Iterated Integrals and Cycles o n Algebraic Manifolds
xii
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The main theorem . . . . . . . . . . . . . . . . . . . . . . .
Appendix: Orientations, Fiber Integration
73 84
95
Bibliography
101
List of Notations
103
Index
107
Chapter 1
Iterated Integrals, Chen’s Flat Connection and m1
1.1
Introduction
Iterated integrals in calculus have the form rx
rt
More generally
Interesting examples: take
dt fl(t)dt = 1-t’
f2dt =
...
n=l
Iterated integrals are related to differential equations, which in turn are related to 7r1, the fundamental group of a manifold: this latter relation involves Lie groups and Lie algebras. So we will begin by proving a theorem of K.T. Chen which describes nilpotent quotients of X I : TI/(TI, TI), ~ 1 / ( ( 7 r 1 ,TI), TI), . . . using certain nilpotent Lie groups. The rest of the book will examine more closely the iterated integrals which give this relationship: “Chen’s flat connection” for 7r1 of compact Kahler manifolds , or more specifically complex projective algebraic manifolds, and find 1
2
Iterated Integrals and Cycles o n Algebraic Manifolds
that we obtain some information on algebraic cycles and on homology of these manifolds. We will introduce a little Hodge theory and the heat operator e - t A , A = Laplacian on forms. The more experienced reader should skip much of the introductory material on Lie theory.
1.2
Differential equations
On the real line R, consider the n x n first order linear system dy' = g(t)A(t) (initial condition y'(0) = E)
dt
(1.3)
y' = ( y l l . . . , y n ) , A ( t ) = given n x n matrix of functions. If A ( t ) is a constant matrix A then the solution is y'(t)= ZeAt. If we take c'= standard basis vector Zi,and y'i(t) = F''exp(At) then we can put these n solution vectors into the n rows of an invertible matrix Y ( t )and rewrite the problem as: find Y ( t )invertible satisfying
d Y ( t ) = Y(t)A(t)dt
(1.4)
Y ( 0 )= I . The solution is given by an infinite series of iterated integrals
+
...+/.../
A(tl)dtlA(t2)dt2* . .A(tn)dtn+ O
The series converges if A ( t )is continuous in t , and we can differentiate term by term to verify (1.4). Next we take X to be any n-dimensional real differentiable manifold, G = G L ( N , R ) ,or any other Lie group, and g = Lie algebra of G, g = M ( N , R ) (all N x N matrices). g can be identified with the tangent space T,(G) a t the identity e ; given C E g we form etc = I tC . ' . , which gives us a path in G (a 1-dimensional Lie group) and C is its tangent vector a t e. Next we define a 1-form a on X with values in g to be a finite sum
+ +
c N
(Y
=
i,j=l
"ij
€4R Cij
Iterated Integrals, Chen's Flat Connection and xi
3
where aij are ordinary (scalar valued) 1-forms on X and Cij are in g. We can in the case where g = M ( N , R ) , regard a as an N x N matrix with 1-form entries. We will write the matrix product of such matrices as a A p. Note that a A a need not be 0 if N > 1. If g is a general Lie algebra we cannot multiply in this way as g is only closed under the [ , ] product. However we define, for a = ai @ Ci, p = pj @ C; (Ci, E g, ai, pj 1-forms E A ' ( X ) ) , the bracket [a,p] as
c;
If g = M ( N ,R) we calculate that a A a = f [a,a ] . (So, [a,a] need not ] [ a ( w ) , P ( v ) ] .We can also be 0). We define [ a , p ] ( vw, ) = [ a ( v ) , P ( w )define da = Cidai @ Ci as a g-valued 2-forms. As a function of pairs v, w of vector fields.
w)= .(.(W))
- w(a(zI))- a ( [ vw , ]).
Here a ( v ) = C a ( v i ) c i is a g-valued function on X , and w ( a ( v ) )is the g-valued function obtained by acting with w (on the scalar function part of a(v)). The main example is the left-invariant Maurer-Cartan 1-form p on a Lie group G with Lie algebra g. Recall first that we can define g = Te(G)and also identify a tangent vector v, E Te(G)with the vector field vg = Lg,(ve) (Lg,=action on tangent vectors of left translation by g, with L g ) . vg is then a left-invariant vector field: Lgl,(v9) = and we have an isomorphism(of Lie algebras) Te(G)4 Lie algebra of all left invariant vector fields, by ve H Lg(ve) = vg. We now define the Te(G)= g-valued 1-form p on G: for g E G, vg E T,(G), @(us)= L9-i+(vg).So if vg = Lg,(v,) then p(vg) = v e . wglg,
Proposition 1.1
The Maurer-Cartan 1-form p satisfies 1 dp f - [ p , p] = 0 2
(us 2-form on G).
Proof: We first evaluate these 2-forms on pairs of left-invariant vector fields v,w. Note that p ( w ) is a constnat (in g) g-valued functions on G, and so v ( a ( w ) )= 0. Thus
4
Iterated Integrals and Cycles o n Algebraic Manifolds
Recall that we defined
(the definition Now
is consistent with and by the definition of u for a left-invariant v, so
and
Thus
for left-invariant vector fields, but any vector field E = C fivi, fi functions and vi left-invariant, and 2-forms are linear over (scalar-valued) functions.
0 Remark 1.1 p is frequently written as the 1-form g-ldg on G. This can be thought about as follows. g denotes the map G -+ G which takes g to g , i.e., the identity map dg is then the differential of the identity map, so for vg E T,, dg(v,) = v,. g-ldg then stands f o r the composite L,-I+ o dg : vg H L,-I*(w,). Using this notation, g-ldg A g-ldg is a matrix product (if G = G L ( N ,R)) and
d(g-ldg) = d(g-') A dg = -g-ldg A g-ldg,
+
so d(g-'dg) g-ldg A g-ldg = 0. W e see now that f o r any map Y : X + G , the pull-back Y ' p = a is a g-valued 1-form on X and satisfies da ; [ ~ , C = Y ]0.
+
Theorem 1.1 If X is a manifold, a a g-valued 1-form o n X satisfying d a $ [a,a] = 0, 20 E X a given point, and G is a Lie group with Lie algebra g, then 1. There exists an open neighborhood U of xo in X and a map Y : U 4 G such that
+
i ) Y * ( p )= a
and
ii) Y ( x o )= e.
2. If U is a connected neighborhood of xo in X and Y : U + G satisfies i ) and ii) of I., then Y is unique and is given by the following formula; if
Iterated Integrals, Chen's Flat Connection and
5
7ri
p : [0,1]4 U is a path from xo to x in U , then if G = G L ( N ,W) c M ( N ,W) or if G is a Lie group in an associative algebra A with identity element e , then
Y ( z )= e
+ l l p * a ( t i )+
11
OStl st251
p * a ( t l ) A p*cx(tz)
+ '.
'
(each term is in A, sum is in G). If p,p' are homotopic paths from xo to x in U ('xed end points during homotopy) then the iterated integral formulas o v e r p and overp' are equal. 3. If X is connected and simply connected, xo E X , then there is a unique Y : X --+ G satisfying i ) and ii) of 1. and given by 2. If X is not assumed simply-connected and 7r : X 3 X is its universal covering with ZOE X , ~ ( 5 0 = ) xo given, then we let ZU on X be n*a. Then there exists p : ( X , f o ) -+ ( G , e ) as above and p gives a homomorphism m : n l ( X , x o ) 4 G such that ? is equivariant. More generally, gives a homomorphism of the fundamental groupoid of X into G . An element of the fundamental groupoid is a path p : [0,1] -+ X with p ( 0 ) = x , p ( l ) = x', up to homotopy. The homomorphism sends p into an element g E G as follows: there is a unique map Yp : [O, I] + G such that Y,*(p) = p * ( a ) on [0,1]and Y p ( 0 ) = e; we let g = Y p ( l ) .I f p : [0,1] -+ Y and p' : [0,1]--+ Y are composable paths, i.e., p ( 1 ) = p'(0) and p i s sent to 9 , p' to g' then the composite path pp' is sent to gg'. Proof: We concentrate on proving the following statement: there is a covering space 7r : X 3 X of X and a map p : --+ G such that p * p = 7r*a and Y ( & ) = e for some point 20 6 X above xo E X . The other statements are easy consequences of covering space theory, while the iterated integral formula is just the 1-dimensional case and has already been discussed. The idea of the proof is to replace maps (defined locally) X + G by their graphs in X x G, and to replace the requirements on the maps by specifications on the tangent spaces to these graphs. Existence of such graphs will follow from the Frobenius integrability theorem. pr3 ( p ) Thus on X x G we form the g-valued 1-form A = prT(-a) (where pri, i = 1 , 2 , are the projections to X , G ) . We write for short, A = -a + p. The tangent space to X x G a t any point (x,g) is denoted T,X @I T g G ,with elements ( v x ,w g ) . Let D,,g c T,X@T,G be the subspace of all (v,, w g )satisfying a(v,) = p ( w g ) , i.e., A ( v Zw , g )= 0. Since p : T g ( G )4 g is an isomorphism, the dimension of D,,g is the same as the dimension of T x X , therefore constant.
+
6
Iterated Integrals and Cycles o n Algebraic Manifolds
In fact Dx,g is the graph of a linear map T x X 4 TgG which pulls back pg to ax. To satisfy the hypotheses of the F’robenius integrability theorem, we have to show that if VI, VZare vector fields on X x G which lie in D x , ga t every (x,g),then [V1,V2]also lies in Dx,g at every ( x , g ) . We can either do this by a direct calculation, or else prove an equivalent condition: the “matrix coefficients” ai of the g-valued 1-form A = -a p , A = C ai @ Ci where the ci are any basis of g, satisfy: each dai E ideal (in the algebra of forms on X x G ) generated by a1 , a2, . . . ; in other words dai = C a j A bji for some 1-forms b j i . To see this we write
+
d A = -da
+ d p = -21( [ a ,a]- [ p ,p ] ) =
1 -2[ ‘ Y - p , a + p ] ,
+
(since [a,p ] = [ p ,a] for any 1-forms a , p ) so d A = -+[A,B ] ,B = a p which implies the condition on the d&i. The Frobenius theorem now says that every (x,g) E X x G has a neighborhood U and a closed submanifold 2 of U , containing (x,g),whose tangent space a t every (x’,g’) E 2 is Now re-topologize X x G so that it becomes a manifold of dimension of dimX with these 2 as open sets. Noting that the projection p r l of Dx,gto T x X is a n isomorphism we see that prl makes the retopologized X x G a covering space of X . (It helps to use the left action of G on X x G ) . Let X be the connected component of ( 5 0 , e) = ZO E X x G. Then X 4X is a covering space and ~ 7 - 2 1 2: X -+ G is a map Y : X 4 G satisfying the required conditions. 0 Exercise 1 Show that in general there is no global map Y : X -+ G with Y * p = a by finding a counterexample - a (very small) manifold X , a 1-form a on it, and a (very small) G , g, such that there is no Y : X -+ G as above.
Question The above proof seems not to have used fully the assumption that G is a Lie group. Can we then state a more general theorem? Prove directly that if Vl,Vz are vector fields on X x G with & ( x , g ) E Dx,gfor every ( x , g ) then [Vl,I41 satisfies the same condition.
Exercise 2
7
Iterated Integrals, Chen's Flat Connection and XI
1.3
Program
We will (following K.T.Chen) construct a special Lie algebra g and a gvalued 1-form 0 ("Chen's connection") for a manifold X - to simplify and make the construction canonical we assume X is compact Kahler. As motivation, recall that for any simplicia1 complex X , the fundamental group can be defined as having generators corresponding to closed edge paths and relations corresponding to 2-dimensional simplices: thus only dimensions 1 and 2 are involved. Here, the Lie algebras 0 will be defined as the free Lie algebra generated by the vector space H1(X; IR) = H I , modulo relations c [HI,H I ]obtained from the reduced diagonal map
A* : H z ( X ; R ) - + H z ( X x X ; I R ) = ( H z @ I R ) @ ( ~ @ . z ) ~ ( H 1 @ -+ Hi @ H I . The image A * ( H z ) c H1@H1 is in fact contained in the skew-symmetric elements C a @ b - b @ a which we identify with [HI,H I ] in the free Lie algebra. There will be a number of technical points involved; g will in general be an infinite dimensional Lie algebra and so 0 will be some kind of infinite series. Similarly its "Lie group" G will be an inverse limit of finite dimensional LIe group. The result will involve also some other associative algebras and another Lie algebra associated to 7r1 = 7rl(X,zO). We refer for the following to [Quillen] and [Lazard]. First, we form the associative algebra R7r1 = group algebra of TI with coefficients IR, then a descendng sequence of 2-sided ideals in R7r1:
I = augmentation ideal > I' > I 3
3
and consider the associated graded algebra Gr(Rr1) = (R7r1/1) @ (1/1') @ . . . @ (In/1"+l)
@.
..
which is furthermore a Hopf algebra. Next we consider the group 7r1 itself and a sequence of normal subgroups (the lower central series ); to indicate groups generated by commutators use round brackets; ( T I , T I ) , ( ( T I , T I ) , 7rl),etc. 7r1 > ( 7 r l , 7 r l ) > ( ( 7 r l , 7 r 1 ) , . i r l ) > . . . > 7 r r n ) > 7 r i n + ' )
>...
where T!') = 7r1 and 7rY") = (7rin),7r1). The quotients 7rin)/7r!"+') are abelian groups and the commutator operation in TI induces a map (a Lie
Iterated Integrals and Cycles on Algebraic Manifolds
8
bracket now)
which makes the associated graded abelian group grr1 = @,"=, 7ry)/7rin+ into a Lie algebra(we will tensor with R). Associated with the Lie algebras g and ( g r r l )@ R are their enveloping associative algebras U ( g ) and U(grr1@R) (also Hopf algebras). The group G associated to g will be contained in a completion, U(g)" of U ( g ) . After all these algebras have been defined we will have: 1) The Chen connection 8, which gives a homomorphism r1 -+ G of groups, will also induce a homomorphism
11 : Gr(Rn1)+ U ( g ) of Hopf algebras. 2) (As shown by Quillen ) for any group (of Hopf algebras)
Q : U(Grn1 @EX)
-+
T I , there
is a homomorphism
Gr(Rr1)
3) Both Q and 11 are isomorphisms
U(Gr(r1)@ R) 3 Gr(Rr1)3 U ( g ) (Quillen showed Q is an isomorphism in general; the proof here will consider only compact Kahler X ) . 1.4
Lie algebras
We begin this program by reviewing the universal enveloping algebra U ( g ) of a Lie algebra g. Analytic definition of U ( g ) where G is a (finite dimensional) Lie group: we recall that the elements of g are first order differential operators on G
in local coordinates which are invariant under left translation. g is closed under [,] but not under usual (associative) product Jv (= second order differential operator). So we consider U ( g ) = the associative algebra of differential operators generated by g= all left invariant differential operators of all orders. However we will give a more algebraic construction of U ( g )
Iterated Integrals, Chen's Flat Connection and
~1
9
(valid over any field); U ( g ) will then be an associative algebra (with unit element 1) containing g (as a sub-Lie algebra) and characterized by the following universal property: given any associative algebra A and any Lie algebra homomorphism h : g -+ A , there is a unique extension of h to an associative algebra homomorphism of U ( g ) to A , agreeing with h on g and taking 1 to 1. To construct U ( g ) we first construct the tensor algebra
on the vector space g, then factor out the 2-sided ideal R c T ( g )generated by all elements of the following form: for z, y E g, r = z 8 y - y 8 z - [z, y] ( E g 8 g @ g) is required to be in R. Thus U ( g ) = T ( g ) / Rand it is easy to see that g 4 T ( g ) U ( g ) is 1-1. The universal property of U ( g ) follows from a universal property of T(g). The Poincark-Birkhoff-Witt theorem describes U (g) as follows: if wl,w2,.. . is any vector space basis of g, then the monomials: 1,zli, wivj(i 5 j ) , . . ' , zli, zliz . . . wik (il 5 . . 5 ik) are a vector space basis for U ( g ) . In this construction, g need not be finite dimensional. In particular we may take g to be the free Lie algebra generated by a vector space V : -+
g=
v @ [V,V ]@ [ [V,V ], V ]@ .
* *
and find that U ( g ) = T ( V )= free associative algebra on V . An important property of U ( g ) is that there is an associative algebra homomorphism
<
Thinking of g as vector fields on a Lie group G, A is the Lie algebra homomorphism induced by the diagonal homomorphism
The associative algebra U ( g ) together with A : U ( g ) -+ U ( g )8 U ( g ) is an example of a Hopf algebra (defined later). The elements of g are primitive
10
Iterated Integrals and Cycles o n Algebraic Manifolds
+ 1C?J z, and in fact
in U ( g ) : A(%)= z @ 1
1 { U E U ( g ) : A(u)= u ( = all primitive elements).
g = P ( U ( g ) )=
+ 1@ u}.
Let I , be an ideal in the Lie algebra g, ie., 5 E I,, y E g +[x,y] E I,. Thus g / I g is again a Lie algebra. Considering I , c g c U ( g ) ,let R be the 2-sided associative algebra ideal in U ( g ) generated by I , (so, R = all C uixivi, Z Avi ~ ,E U ( g ) ,z i E I,). Then U(g/I,) is naturally isomorphic to U ( g ) / R . (Use the universal property).
Exercise 3
In the Chen description of n1( X ,5 0 )we will use the following Lie algebra g: consider the free Lie algebra L generated by H l ( X ; R ) namely, L = HI @ [HI,HI] @ . . . . Note that [H1,H1]= {CaC?J b - b @ a : a , b E H I } . Furthermore consider the diagonal map A : X -+ X x X and the following sequence of R-linear maps
The image of A, actually lies in [HI,H I ]=the skew-symmetric elements of H I @H1 (dually, the cup product map H1@ H1 + H 2 is skew-symmetric). Let now I be the Lie algebra ideal in L(= free Lie algebra on H I ) generated by A*(Hz(X;R)). We define g = L/I=Lie algebra over R. The Hopf algebras we will use are associative algebras H over R with unit element 1 and algebra homomorphism E : H + IR ( ~ ( 1=) 1) so that ker E, denoted His a %sided ideal and H = IR1@H as R-vector space. The Hopf algebra structure is an associative algebra homomorphism
A :H
+H @
H,
A(1) = 1 @ 1,
such that if h E H then
A ( h )= h @ 1
+ 1@ h + Chi @ hy
(hi,hy E H ) .
Thus
=m.
A ( H )c ( H @ R l ) $ ( R l @ H ) @ ( H @ H )
The main examples of Hopf algebras we use are: 1. H = IRn, 7r any group. (In fact, n could be just a semigroup with identity e, i.e., a “monoid”). A is just the homomorphism
A : Rn + R(n x n) = IR7r @ Rn
Iterated Integrals, Chen's Flat Connection and
given by the diagonal homomorphism A : IT
4
IT
A]
11
x IT,
2. Similarly, U ( g ) , for a Lie algebra g has diagonal map A : U ( g ) 4 U ( g )@ U ( g ) given by the diagonal homomorphism
B-+fl@B,
Z H (z,z).
+
So, A(z) = z @ 1 1 @ z for z E g. 3. If H is a Hopf algebra with map A , a 2-sided associative algebra ideal I c H is called a Hopf ideal if
A(I) c I @ H
+ H @ I c H @H.
Then H / I is again a Hopf algebra (with diagonal map induced by A). Note that for all n 2 1, A(I") c H @ I n
+ I @ In-' + . . . + I n @ H = A ( I ) n .
Thus the associated graded associative algebra
@n>OIn/In+l =H/I
@ 1/12 @
...
is also a Hopf algebra if I is a Hopf ideal.
Definition 1.1
1. An element z E H is called primitive if
A(z) = ~ @ l + l @ z . The primitive elements are a Lie algebra under [,I, denoted P ( H ) . 2. An element g E H will be called "group-like'' if it is invertible and satisfies A(g) = g @ g. The group-like elements are a group under multiplication. Note that g = 1
+ i j , ij E H , if g is group-like.
Example 1.1 a ) The 2-sided ideal I in H generated by a collection of primitive elements z is a Hopf ideal. b) The primitive elements in U ( g ) are exactly g (use the PoincarkBirkhoff-Witt theorem to prove this!) c) The group-like elements in RIT are just IT (for T a group). (Prove this!)
12
Iterated Integrals and Cycles o n Algebraic Manifolds
We look now at the process of completing certain Lie algebras and their enveloping algebras, to obtain something resembling formal power series, in which we can construct exponential and log functions.(See [Lazard]). First, let L be the free Lie algebra on a vector space V (in our applications V will be finite dimensional). Then U ( L )is the tensor algebra (or free associative algebra) T ( V ) and L is the Lie subalgebra of T ( V )generated by v;
where the subspaces L1 = V , Li+l = [Li,V ] satisfy [Li,L j ] C Li+j. L is called a graded Lie algebra. For each n, $i>nLi denoted L>, - is a Lie ideal. Thus we have a sequence of onto homomorphism of Lie algebras ---t
L / L l n 4 L/LLn-l
--+
.
"
--f
L/L>2 = v
+0
and the inverse limit is a Lie algebra L A whose elements may be written as infinite series
Next consider a graded ideal I L in L ;
I L = ( I L n L1) EB ( I Ln L 2 )EB . . . . Then L / I L is again a graded Lie algebra g and we can form the completion GIA;
g = g 1 $ g 2 CB ... , gi = L i / I L n Li and gA = l i m ( g / g > n )
+ +
with elements Z" = z1 z2 . . . (again formal infinite series). Since g = L / I L , U ( g ) = U ( L ) / R ,R= %sided ideal in U ( L ) generated by I L . So R is again a graded, or homogeneous ideal and U ( g ) = U ( L ) / Ris a graded associative algebra generated as associative algebra by its degree 1 elements g 1 = V / ( I L n V ) . For simplicity we will assume I L n V = (0) so g1 = V = L1 (as will be the case in our examples). Now consider in U ( g ) the ideal of all elements of degree 2 n, which is just (Vl)" = (g)" = U ( g ) l n (note this is larger than the ideal generated by
Iterated Integrals, Chen's Flat Connection and
x1
13
g2n). Then the inverse limit of the U(g)/U(g)>,,will be denoted U(5)": it is an associative algebra with diagonal map homomorphism L
Since (Vl)" n g = g>,,, - g" embeds in U(g)" as Lie subalgebra: both of these have elements which are all the infinite series, with terms in gi or v,Z respectively. 1. The primitive elements of U(g)" with respect to A" are just the elements of g". 2. Let x E U(g)" have constant term 0, and define exp x = 1+x+x2/2!+ . . . , an element o f U ( g ) " . Similarly define log(l+x) = x - x 2 / 2 + x 3 / 3 - . . . . Then: a ) if p E U(g)" is primitive, then exp(p) = g E U(g)" is group-like:
Theorem 1.2
A*(exp P) = exp P c 3 exp P E ( U ( 8 )8 U ( g ) ) " (and expp has constant term 1 and is invertible). b) If g = 1 x E U(g)" is group-like then logg is primitive (and so is in 0"). 3. We have bijections which are inverse to each other
+
exp
Primitives
__f
= g"
1%
Proof: 1. If p = pl +pa
+
*.
G = group-like elements of U ( g ) " .
. , (pi E U ( g ) i then
AYP)
= PeD 1
+ 1eDp
if and only if
for each i, so pi E gi (primitives of U ( 5 )= g). 2. a ) A" (p) = p 8 1 1 @ p implies
+
+
A"(expp) = exp(A"(p)) = exp(p 1 1 8 p ) = e x p ( p 8 1)exp(1eD p) as p 8 1,18p commute = expp 8 expp,
so expp = g is group-like.
Iterated Integrals and Cycles o n Algebraic Manifolds
14
b) If A''(g) = g @ g (g = 1
+ x) then
so logg is primitive. 0
In the discussion above we can replace g by g / g l n which is an ( n - 1) step nilpotent and graded Lie algebra. Then in the completion U(g/g2n)" we have = g / g 2 n (each series actually has only a finite number < n of terms) and the corresponding group exp(g/gln) has multiplication defined purely by the bracket in exp 2 exp y
= exp(a:
1 + y + z[x, y ] + . . . ).
Suppose further that g is finite dimensional over R: then - is also finite dimensional and the corresponding group is diffeomorphic to g / g > n via exp, log and is a nilpotent Lie group.
1.5
Chen's Lie algebra and connection
Now let us consider a connected manifold X with finite dimensional Nl(X;IW) = H1 and define g as (free Lie algebra on H I ) / ideal (&H2). Let us consider any g'-valued 1-form 6 on X , by which we mean an infinite series
e=e1+e2+ with
ei
=1-form on X with values in g. We will assume that 1 de -t 2
-[e,el = 0,
and that 81 has the following special form: choose any basis h1,i of H1 over let at be closed 1-forms on X whose cohomology classes are dual to the h l $ i ,i.e.,
R,and
Then we assume
6'1
=
xiat
@ h1,i.
Iterated Integrals, Chen's Flat Connection and xi
15
If we fix n > 1 and consider g / g l n = g ( n ) , we get a "homomorphic image" O(n) = 6 mod g2n which again is a flat connection and gives a homomorphism which we denote
II
: 7r1(X,20) -+ G(n) = Lie
group of g / g r n
explicitly given by the iterated integral series whose degree 1 term, for an element y E nl(X,ao) is just Ci(&ai)hl,i = image of y in H,(X;R) = m / [ T , m €9R ] The group homomorphism given by O(n): h(n) : ~ 1 (2x0,)
+
G(n) c U ( B / B ~ ~ ) "
(also denoted I I ) extends to an associative algebra homomorphism, again denoted generically as I I I b l
+
U(B(n))"
which is compatible with the diagonal maps A , A' since group elements in 7r1 go into group-like elements in G(n): it is a "Hopf homomorphism" h ( n ) , which is compatible also with homomorphisms given by decreasing n. Thus, h(n)commutes with homomorphisms to R defining the augmentation ideals, and so induces also a homomorphism of the completion (R7r1)" + U(g(n))" and a homomorphism of the associated graded algebras:
Grh(n) : Gr(Rr1)
-+
Gr(U(B(n))A)= U ( g ( n ) )
(since U ( g ( n ) )is a graded algebra). This last homomorphism induces just the natural isomorphism on the degree 1 elements
(17rl)/(17rl)2 = Tl/(Tl,Tl) €9R -+ H l ( X , R ) = g 1 which are generators, and so Grh(,) is surjective. The Grh(,) for increasing n are compatible and define a Hopf homomorphism(surjective)
I I = Grh : GrR7r1 4 U ( g ) . We also have surjective homomorphisms (R7r1)' -+ U(g)" and U ( ~ ) / ( Bfor) ~all n. We will prove all of these are isomorphisms.
R 7 r 1 / ( 1 ~ 14 )~
Iterated Integrals and Cycles o n Algebraic Manifolds
16
Before proving this, we want to construct connections 9 satisfying the two conditions above. We will consider only compact oriented (connected) manifolds X satisfying a special condition on 1-forms and 2-forms, which will be satisfied by all compact complex Kahler manifolds (and in particular by all non-singular complex algebraic varieties in projective space). The assumption is that we are given R-linear subspaces:
XI, C’of A1(X),
X2,E20f A2(X) (Ai(X) = all real valued differentiable i-forms) satisfying
1) !Xi C kerd and Xa
--+
kerd/dAi-’ = HhR(X) is an isomorphism.
2) d : A’ 4 A2 induces an isomorphism C’ 4 E2 3a) %’ A X ’ C X2 + E 2 (3a just says that if a ,/3 E X’ so that a A /3 represents a cohomology class in H 2 , then a A p = y q, where y E X2represents this cohomology class and E E2 c dA’ so q is exact).
+
3b) [ ( X ’ + C ’ ) A C ’ ] n k e r d c E2. We remark that 1, 2, 3a hold on any Riemannian (compact) manifold if we write W= harmonic pforms
AP = XP @ dAP-’
@
d*AP+]
and take C1 = d*A2 ( = “coexact” 1-forms) ,
E 2 = dA’ = dC’ = exact 2-forms.
However 3b requires that the metric be Kahler as well (we define this later). A basic theorem of K.T.Chen is: Theorem 1.3
(K. T . Chen) Given the subspaces
XI, C1 c A1(X); X2,E2 c A2(X) satisfying 1, 2, 3 above there is a unique
B
= B1
+Bz
+ .. .
Iterated Integrals, Chen’s Flat Connection and xi
17
(formal infinite series with 8i E A1(X)@J.J~),where J.J is the free Lie algebra on HI, modulo relations A,(H2), satisfying the following 3 conditions I. 81 E 3C18 HI and represents the identity map HI +. H1 if we identify X1 with H1 = vector space dual of H I . ( W e are assuming these spaces are finite-dimensional): 81 = C Qi 8 xi.
III. de
+ +[elel = 0.
Proof: Existence is shown by constructing 81,132,.. . inductively, starting with 81; the proof will also give uniqueness. So we start with any basis x1,x2,.-. of H1 = 81 and “dual” basis a1,a2,... of X1, i.e., 23 ayi = & , j , and take 81 = ai 8 x i . Thus, dB1 = 0, while I11 requires dB2 = -4[81,191]. Here
xi
By 3a), ai A
a? = %(ai A a j ) + aij
where %(ai A a j ) E X 2 and aij E E 2 ;these last two elements are uniquely determined by ai A aj and aij = -aji since a i A aj = -aj A ai. By 2), d : C1 -+ E 2 being an isomorphism, there exists a unique aij E C’ with
so 03% . . - -a.. 23’ Now the requirement 111 says that 1 de2 = --[el,el1 2
We will show that
K(ai A a j ) 8 [xi,x j ] = 0 in A2 8 8 2 so that the above
Iterated Integrals and Cycles on Algebraic Manifolds
18
equation reduces to
Since d : C1 ---f E2 is an isomorphism, this equation for solution in C1@ gz, namely
02
has a unique
Now we show that for the basis ai of X1 dual to basis xi of Hl we have
C X ( a i A aj)8 [xi,
zjl=
o E 3~’ CZJ8 2
( X ( a iA a j ) = harmonic part of cri A a j ) : the left hand side can be regarded as a linear transformation HZ --+ gz (identifying X 2 and H 2 ) which takes a homology class TZ E H2 to ( 2 2 , ai A a j )\xi, x j ]. (( z2, ai A a j ) means Jz2 ai A aj = J,, X ( a i A aj).) It is more convenient now to write
ci,j
and show
. if and we have to show that x i , j ( a i 8 a j ) 8 z i z jvanishes on all A , ( z ~ ) But we consider zizj as being the image of z i @ z j ,which is in the free associative algebra on H1 , then C(ai 8 a j ) 8 (xi 8 zj) is the linear transformations of H1 8 H I to itself taking h 8 h’ to C ai(h)zi8 aj(h’)xj = h 8 h’, ie., the identity linear transformation. Thus C(ai8 a j ) 8 (xi 8 zj) takes A,(+) to itself and C ai 8 aj 8 X i X j takes 6 , ( z 2 ) to its image in 8 2 , which was expressly designed to be 0. Now for On, n 1 3, we already have 01 E 2-C1 8 01 and we assume we have also defined Oi E C 18 gi for 1 < i < n satisfying: for 1 5 j 5 n - 1,
Iterated Integrals, Chen's Flat Connection and a1
19
Consider n-1
18i
A Bn-i
E
( X 1+ C ' ) A C1 8 gn.
i=l We want to show that d(CY1: 8i A On-i) = 0. Then assumption 3b will give us that the 2-form coefficients will be in ((XI C1) A C') n ker d = E 2 , and E2 is isomorphic to C1 via d , so there will exist a unique 8, E C' 8 gn such that den = 8i A On-i which satisfies condition I11 for On. But,
+
zyI:
= a sum of terms f8, A 8 b A gc.
Consider triples ( p , q , r ) with p , q , r 2 1 and p + q + r = n. Then 8, A Oq A 8, occurs in this sum: once with coefficient -1, in (do,+,) A 8, once with coefficient +1, in -8, A doq+, and nowhere else. So the sum is 0 concluding the proof of the theorem.0 We come back to our program of constructing homomorphisms of Lie algebras, associative algebras, and Hopf algebras which we will show are isomorphisms. First of all, from the connection 8 we have homomorphisms, denoted II(=iterated integral) of groups: 7r1 --+
G = group-like elements in U(8)''
of Hopf algebras:
of graded algebras:
1.6 Some work of Quillen Next, following Quillen and Lazard, for any group 7r we consider its descending central series and associated graded abelian group Grn, which is a Lie algebra (using group commutator in n).
20
Iterated Integrals and Cycles o n Algebraic Manifolds
s,
The function q : 7r 4 q(y) = y - 1 induces a homomorphism Q of a Lie algebra to an associative algebra
Gr7r t GrR7r and so a homomorphism
Q : U(Gr7r8 R) -+Gr(R7r) of Hopf algebras. We will now give the details of Q. We will use the following construction studied in [Quillen]: for any group 7 r , form the descending central series ((, ) denoting group commutator) 7.p) = 7r
= (r, ).
3
J
. . . 2 n(n+l) = (&I,
).
J -
and consider the graded abelian group Gr(7r) = algebra using group commutator in T (see [Lazard]). We want to construct a Lie algebra homomorphism
as a Lie
(Gr7r)8 R -+ Gr(&) = $ n >-l ( & ) n / ( & ) n f l , where [, 1 in Gr(%) is given by commutator in this associative algebra. We start with the function q(y)=y-1
q:7r+&,
inducing the abelian group homomorphism 7r(1)/7r(2) (Y7r(2))
4
rwn/(&)2
4
q(y)
+ (Ey.
This induces an isomorphism over R: 7r(1)/7r(2)
8R
4
&/(&)2.
These identities will be used:
+ ( 7 2 - 1)+ (71- I)(%- I),
a ) 4(YlYZ) = 7 1 7 2 - 1 = (71 - 1) b) -1
dYlY2Y1
-1 72
-
) - ((71 =
= elements =
- 1) - (72 - l)(n-
1))r;'r;l
[(n - I), ( 7 2 - 1)](1+(rl1r;l - 1) of
[(n- I), ( 7 2 - l)]
mod 13.
Iterated Integrals, Chen's Flat Connection and
~1
Next, assume inductively that for a given n 2 2, q ( x p ) ) E dn-'), 7 2 E 7r then
21
and if
y1 E
Q((Y1,72)) =
[ q ( n ) , 4(^12)1 mod
( W n + l ) .
Using u ) and then b) we prove that ----71
q : TI"' --i Rx
-+I
/Rx
is a group homomorphism, induces q : 7p/x;"+l)
-+I Rx /R7r I I
--i
TI"),
also a homomorphism, and if y1 E 72 E x then q ( ( n , y 2 ) )belongs -+2 to Thus q(7ry")) c and = [yl- l , y 2 - 11 mod Rx ----n ----n+l q induces 7rin)/xin+') 4R7r /R7r and takes group commutator (,) E -n+l ----n+2 ( 7 r p ) , x 1 )into algebra commutator [,I E Rx . Finally, one can /Rn show that q induces a graded Lie algebra homomorphism Gr(7r)4 Gr(Rn) - -2 which is an isomorphism on x / d 2 ) @I R + R7r/Rx . Thus one gets a homomorphism of associative algebras and even of Hopf algebras
sf'
.
Q : U((Gr7r)@I R) + Gr(R7r). Also (see [Quillen]), Gr(Rn) is the universal enveloping algebra U(P(GrRr)), where P denotes the Lie algebra of all primitive elements (the Hopf structure in Gr(R7r) arises from that in Rx). The Hopf algebra map Q thus induces a Lie algebra map Q : (Grx)@I R 4P(GrR7r). Similarly I I induces Lie algebra map
I1 : P ( G r ( R x ) )--i g = Gr(8'')
= P(U(g)).
We aim t o show that all these Q , I1 are isomorphisms.
1.7 Group homology We will use some homology theory of discrete groups x = 7r1 (X). Let B7r be the classifying space of x; it is a K ( x ,1) space, i.e., its fundamental group is x and its higher homotopy groups xn for n 2 2 vanish. The homology
22
Iterated Integrals and Cycles o n Algebraic Manifolds
groups with coefficients in any Rx module M , Hi(B7r,M) are denoted Hi(n,M ) (and depend only on x). For i = 0, Ho(x,M ) = M / T G M . For the (connected) manifold X and x = 7r1(X1xo),we construct Y = Bxl as Y = XU cells of dimension 2 3. Thus the inclusion X -+ Y induces an isomorphism 7rl(X) 4 x1(Y). Up to homotopy we can replace the inclusion X c Y by a fibration p : X -+ Y , with X -+ X a homotopy equivalence, with fiber p-l(y0) = F (where yo= base point of Y ) . Notationally we will write X instead of X (we know H i ( X ) = Hi(*) and 7rz(X) = .Z(X).) For any fibration F 4 X -+ Y as above (a “Serre fibration”), we have some exact sequences of homology groups, called exact sequences of low order terms in a spectral sequence: see H. Cartan and S. Eilenberg’s book “Homological Algebra” (p. 328 Th.5.12a with n = 1). In these H I (F) = H I ( F ;R) is a 7r1 = x1 ( X ,5 0 ) module and H I (F),l means
Hl(F)/&Hl(F). The exact sequences are
Hz(X)4 H z ( Y )
+
Hi(F),,
+
Hi(X) 4 H i ( Y )
0.
(1.7)
Hz(X) H z ( Y ) 4 0
(1.8)
-+
If further H1(F) = 0 then
Hs(X)-+ H 3 ( Y )
-+
Hz(F),,
-+
-+
is exact. Further, there is the long exact sequence of homotopy groups (for all i 2 0);
. . . 4 Ti+l(Y)-+ T i p ) 4 T i ( X )
-+
xz(Y)4 . . . .
7ri
(1.9)
We have assumed xi+l(Y) = 0 for i L 1 and xl(X)4 xl(Y)is an isomorphism, so xz(F) 1x z ( X ) (- means isomorphism). and xl(F)= (0). The Hurewicz theorem now gives an isomorphism 7rz(F) H2(F) and a commutative diagram
m(F)
-=-+Hz(F)
1
-1
Thus image of H z ( F ) -+ H z ( X ) ( = image of Hz(F)/&Hz(F) = image
-+
Hz(X))
x z ( X )4 HZ(X) spherical
= (definition) : H ,
(X)= Hz”Ph(X),
Iterated Integrals, Chen's Flat Connection and
23
~1
Now (1.7) and (1.8) together become H 3 ( X ) -+ H 3 ( Y ) --* H2(F),,
-+
H 2 ( X ) -+.H2(Y)
+
0
---*
HIP)
f f l ( Y )+ 0.
So we get a short exact sequence 0 -+ H2"Ph(X)+ & ( X )
due to Hopf. Now we apply A, : H2
-+
4
H 2 ( Y ) -+ 0
H I €4 H I , to the various spaces, and note that
6 * H 2 ( S 2 )= 0 as H l ( S 2 )= 0.
So, A , H l p h ( X ) = 0 , giving a diagram 0
-
Hz"Ph(X)
1 0
-
H2(X)
1
lT*
-
HdY)
I
-
0
x*
H l ( X ) 63 H l ( X ) A H l ( Y ) 63 H l ( Y ) .
So image 6, c H 1 ( X ) @ H 1 ( X can ) be identified with image A, C H l ( Y ) @ Hl(Y), y = qm(x)). Next we look at spaces Y = B(group): for any normal subgroup N of a group (discrete), we have a fibration
We write Hi(B(group);Z) = Hi(group; Z). Then the exact sequence (1.7) becomes, with coefficients Z,
Here
Iterated Integrals and Cycles on Algebraic Manafolds
24
Using this in (l.lO), we get a sequence with exact rows and diagonals 0
0
I
T N / N n (I-,r)
T
~ ~ p - H7 +
~
~
N
+
N/F, N) )
I W,r) ( r / ~ ) / (r r// ~~ )0 -+
-+
--f
I1
r/N(r,r) NOW take r = n l ( x ) = T , N = (n, n),r / N . = nab, N/(r, N ) = nf?)/ny).The exact sequence ( l . l O ) , with coefficients in R (but Q would also work) gives the exact upper row of the following diagram, whose lower row is exact by definition:
I (Hl(T1)= H l ( X ) = Hl(Y) = H1).
Lemma 1.1
”;
~ 2 ( n : b ; ~ )~ l ( n r CQR ~ ; H1(TTb;R)skew ~ )
is an isomorphism.
Proof: We may assume n1 is finitely generated, then pass to direct limit. So nyb is also finitely generated. Now nyb = ZT @ Torsion so H2(nrb;IR) 2 H2(ZT;R). Since B ( Z ) = S 1 , B ( Z T )= S1 x ... x S1(r times) =Tr: r-dimensional torus. Thus, H i ( T T ) = A i ( H 1 ( T r ) ) . Passing to the dual vector space H i ( T T )we see that &*=cup product: A 2 ( H 1 ( T r ) )-+ H2(T‘) is an isomorphism. 0
So in the diagram (1.11) preceding this lemma the vertical A, is an isomorphism, and we conclude that the last vertical arrow is also an isomorphism(induced by A, on H2(nyb)) (n1( 2 )/n1(3) ) @
+
A2(H1)/A*(H2).
Iterated Integrals, Chen’s Flat Connection and
TI
25
(Hi are Hi(X;R), x1 = x l ( X ) We now conclude: (xl(2) /xl )@R = Gr(x1)(2)has the same R-dimension
h
as A 2 ( H i ( ~ ) ) / & H z (=~8)2 . 1.8
The basic isomorphisms
As consequence of this calculation, we have: Assuming x1 ( X ) a b= H I ( X ) is finitely generated, the previously defined surjective linear map
II
0
Q :Gr(~i)(z)
-+
gz
is an isomorphism. We previously knew that
II
0
Q : Gr(ri)(i)= xi xi, xi)
+
81 = H 1
is an isomorphism (natural isomorphism), and induces I I o Q in degree 2 (as deg 1 elements generate deg 2 by commutator). Finally, we recall that g is the free Lie algebra L(H1) on H I modulo the relations &,(Hz) contained in [HI,H i ] . So the “identity map” Hi -+ (Grxl)(l) extends to a Lie algebra homomorphism cp : L(H1) -+ (Grx1)@ R = Gr .
We also have the natural quotient (onto) homomorphism $J : L(H1) -+ g, all of these fitting into a triangle of Lie algebra homomorphisms, which commutes since it commutes on generators=degree 1 elements:
L( HI) -5 Gr(x1) @ R = G r
JII~Q
$\ g
In degree 2, 11 o Q is an isomorphism and so ker$Jz = A*Hz = kervz. Since q(&H2) = 0, cp induces a Lie algebra homomorphism g 4 Gr(x1) @ R, which is inverse to I I o Q. Thus
Theorem 1.4 I I o Q , 11, and Q are all Lie algebra isomorphisms (of the associated graded algebras):
Q : (Grxl) @ R -+ P(Gr(&)) I I : P(Gr(&))
+g
= L(H1)/(A*H2)
26
Iterated Integrals and Cycles on Algebraic Manifolds
and so also isomorphisms of the universal enveloping algebras
Q : U((Gr7r1)@ R) + Gr(R7rl)= U(P(GrR7r)) 11 : Gr(R7r1)
---f
U ( g ) = Gr(U(g)”).
Finally, since 11 arises from a (Hopf) algebra homomorphism R7r1 -+ U(g)”, and is an isomorphism on associated graded algebras, it is also an isomorphism on completions:
11 : (R7rl)A
---f
U(g)A
is an isomorphism. 1.9
Lattices in nilpotent Lie groups
The term “lattice” here means a subgroup D of a (connected, simplyconnected) nilpotent Lie group N such that D is discrete in N and N / D is compact.
Theorem 1.5 Assume 7 r l ( X , x o ) i s finitely generated. Let G(,) = exp(g/g2n) = exp(g(,)) as before, and 11 : 7 r l ( X ; z o ) + G G(nl the monodromy map given by iterated integration of Chen’s connection 8. Then, the image of 7rl(X,zo) in G(,) is discrete and co-compact; i.e., G(,)/(image) is a compact manifold, for each n L 2. Further, this image is just 7rl(X,20) modulo and torsion elements mod 7rr”’). ---f
(.in’
Proof: We use induction on n and the following lemma:
Let G be any finite dimensional connected Lie group and N a connected closed normal subgroup of G (which we may assume is also a Lie group), GIN = H the quotient Lie group. Suppose D C G is a subgroup such that D n N is a discrete co-compact subgroup of N , and DID n N c GIN is also discrete co-compact. Then D c G is discrete co-compact.
Proof of Lemma We consider the principal fiber bundle N f G 1:GIN. Since D l D n N is discrete in G I N , there is a neighborhood U of the identity eN in GIN such that U n ( D N I N ) = eN. If U is small enough, there
Iterated Integrals, Chen's Flat Connection and
AI
27
is a section s : U --t p-'(U), s ( e N ) = e. Then p - ' ( U ) = s(CI)N is diffeomorphic to U x N (by multiplication) and D n s ( U ) N = D n N . Since D n N is discrete in N , there is a neighborhood V of e in N so that D n V = (e). Now s ( U ) V is a neighborhood of e in G and D n s ( U ) V = ( D n N ) n V = D n V = ( e ) , so D is discrete in N . To see G / D is compact, note that GIN is the union of the D I D n N translates of finitely many compact sets, and N is the union of the D n N translates of finitely many compact sets. 0 Next we prove the theorem. Recall our notation: gqn) = g / g ~ G(nl ~ , = e x p q n ) Furthermore g n = [
gin] = g 2 n , and g(") = g/g["l, G(n)= G/G(") (G(n)denoting the iterated commutator subgroup). We now have two exact sequences of groups related by the group homomorphisms I 1 : n + G(n). [ [ g l l g l ] , . . .,611 (n factors) denoted gyl. And so
(1)
(1)
-
-
G(")/G("+l) TI1
7r(4/7r(n+1)
__f
G(,+I)
T
Ir
7r/n(n+l)
- TI1 __f
G(n)
(1)
7r/n(n)
(1).
We need to prove that the leftmost vertical arrow is an isomorphism when we tensor ~ ( ~ ) / 7 r ( ~ +with ' ) R. To see this we show that the following diagram commutes (all maps are group homomorphisms as the groups and Lie algebras are abelian):
Here GrII is the isomorphism GrW7r -+ U ( g ) , Prim denotes primitive elements (which map isomorphically to 0). I I ( y ) E G(") since I I is a group homomorphism, and For y E dn), Gcn) = expg["l, so
with
Iterated Integrals and Cycles on Algebraic Manifolds
28
On the other hand,
Q ( y ~ ( ~ +=l (y ) ) - 1)
mod
(G)"+l,
and
G T I I ( Q ( ~ ~= ~ I+I~( )-~)1) = I I ( ~-) 1 = x, again.
mod g["+'l
We conclude by recalling that G r I I is an isomorphism for each n. 0
1.10
Some Hodge theory
We have still to construct the differential forms 3c1, C1,3c2,E 2 used in the connection 0 and for this purpose we will give an introduction to Hodge theory on Riemannian and Kahlerian manifolds, then define these 1- and 2-forms. This will complete the proofs of the discussion of 7r1 and nilpotent Lie groups.
A. Compact Riemannian and Kahlerian Manifolds 1. Riemannian manifolds X . Start with X any C" manifold of dimension n , A P ( X ) = all C" pforms. We assume that for each x E X with T,X the tangent space, we are given a positive definite symmetric bilinear form g z = g : T,X x T z X -+ W. g gives an isomorphism G : T, -+ T: = cotangent space, by G ( v ) ( w )= g ( v ,w) and so we can define g as a bilinear form T * x T* + W, g ( G ( v ) ,G ( w ) ) = g ( v,w). We assume g is differentiable with respect to x. For each integer p , 0 5 p 5 n = dimX we can define g 1 APT* @APT* -+ R,again positive definite symmetric, by
g ( l l A . . . A ,Z 1'1 A . . . A Zb) = det[g(li,1;)]. Next we assume X is oriented, i.e., we have a (continuous) choice of one of the two connected components of (AnT:) - (0), called the "orientation class".
Definition 1.2 The volume element dvol E AnT: is the unique element in the orientation class satisfying g(dvo1,dvol) = 1. If e1,. . . en is an orthonormal basis of T* with e1 A tation class, then this n-covector is dvol.
. - .en in the orien-
Iterated Integrals, Chen's Flat Connection and
29
~1
Next we can define the Hodge +-operator APT: An-PT:, a linear isomorphism. First we recall the interior product operator -+
il : APT*-+ AP-'T*, for 1 E T* given by il(l1 A
. . . A l p ) = g(1,11)12 A -9(1,
/2)11
* * '
A
1,
A 13 A
. . . A 1,
' *
.
+ ( - l ) p - l g ( l , l p ) ~ ~A . * * A l p - l . Now we define * as follows: for *(I1 A
A main property of
3
* *
11
A
. . . A 1,
A l p ) = ilpilp-l *
*
E
A" T * ,let
. ill (dvol).
* is: for a,P both E APT*,
a A *P
= g ( a , @duo1 = p A
*a.
Also, * : AP -+ An-P, * : An-p -+ AP and -M( = ( - l ) P ( " - P ) on AP. We may let the point 2 vary in the above constructions which take place for each T:, and obtain the operator * taking pforms Ap(X) to n - p forms An-P(X) (all forms are real valued). Thus, for a,P E AP(X),g ( a , P ) is a Co3function and g(a,P)dvol = CY A *P E A n ( X ) . Also, g ( a , a ) 2 0 and g(a,a ) = 0 implies a = 0. Suppose further X is compact. Then each Ap(X) is a pre-Hilbert space with inner product
*D Next we bring in the exterior differential d , and define an operator
d*=f*d*:Ap+'-+Ap so that for a E AP(X),y E Ap+l(X),
(da,Y)X = ( a ,d * y ) x .
The exact sign in d* = Ifr * d* is determined by using - -1 P ( n - P ) a P *n-p(*paP ) ( )
and Stokes's theorem. For n even d* = - *d*.
Iterated Integrals and Cycles o n Algebraic Manafolds
30
The Laplace operator A : A P ( X ) + A p ( X ) is defined as A = dd* id*d.
Definition 1.3
3CP = kerA = { h E A
p : dd*h
+ d * d h = 0).
Equivalently, 3P = { h E A P : d h = 0 and d * h = 0). It is easy to calculate that the three subspaces XP, d*Ap+l =image d*, dAp-' = image d are mutually orthogonal. We will assume known (or accepted) the following basic theorem from elliptic PDE.
Theorem 1.6 Let X be compact oriented Riemannian and let ,8 E A P be orthogonal t o W ; that is, for all h E 3P, /3 A *h = 0. T h e n there is a unique a E AP such that
sx
A a = 0.
Further, 3l
= ker A
is finite dimensional.
Assuming this theorem, we prove that AP is the orthogonal direct sum of the three subspaces 3P, dAp-', d*Ap+l:
AP = XP
@
dAp-l
@
d*Ap+l
(Hodge decomposition).
To see this, start with any y E AP and let
/3 = y - (orthogonal projection of y on 3-c) i
where hi are an orthonormal basis of 3P. By the theorem, there is a unique a so that
so
y = P(y)
+ d(d*a) + d*(da).
Next we show that d : d*Ap+l -+ d A p is an isomorphism, and d* : d A P -+ d * A p f l is an isomorphism. To show d is onto, we just note that d(3-cP) = 0, d(dAp-') = 0 so dAP = d(d*AP+'). To show d is 1-1on d*Ap+l, let y E d*Ap+' and d y = 0. Then d*y = 0 as d*d* = 0 (d*d* = ( d d ) * ) . Now dy = 0, d*y = 0 implies y E X,but 3c and d * A p f l are orthogonal, so y = 0. Similarly d* is an isomorphism.
Iterated Integrals, Chen’s Flat Connection and x i
31
We conclude that ker d = 3cP @ dAp-’ and so 3cP -.+ kerd -+ kerd/Imd = H g R ( X ) is an isomorphism. In other words, the natural onto map kerd 4 kerd/Imd = HP is “split” by the choice of a complement to Imd in ker d, namely the orthogonal complement XP. The Hodge * operator sends 3cP isomorphically t o Xn-P, dAp-’ to d*An-Pfl and d*AP+l to dAn-P+1. Note that * does not behave simply with respect t o wedge products or d , and in general the wedge product of two elements of X need not lie in X:if Q,P E X then Q A ,B = y dq, y E X,d v E dA.
+
B. Complex manifolds, Kahler manifolds First consider C” with complex valued coordinate fuctions z j = x ~+j i y j . By tangent space at a point p we mean the tangent space of the underlying real manifold with real coordinates 21, y1, . . . , x,, y,. Thus, Tp has R-basis
a
a d ... -
-
a -
ax, ’ By,
aX1’ayl’
’
We define the (R-linear) operator J : Tp -+Tp by
a i = l , . . . n. dyi axi axi dyi (so Tp is n-dimensional over the field R + RJ isomorphic to C). a
J(-)
a
= -,
a
J(-)=--,
On the dual vector space Tp*with dual basis dxl, d y l , . . . , dx,, dy,, the transpose of J, denoted J again, acts by J(dyi) = dxi,
J ( d ~ i= ) -dyi.
Let U , V be open subsets of C” and F : U -i V a CO” map. We say that F is holomorphic if the differential d F : T p ( U )4 T F ~ ~ , satisfies (V) dFoJ= JodF
(allpEU).
These are the Cauchy-Riemann equations. We can now define a complex manifold X to be a real 2n dimensional differentiable manifold with linear automorphisms J of T p ( X )for each p, with J2 = -Id, such that the coordinate charts F taking open sets U in Cn to open sets V in X satisfy d F o J = J o d F (and so the “change of coordinates” FC1 o Fz, mapping an open set in Cn to another open set in Cn, are holomorphic.)
32
Iterated Integrals and Cycles on Algebraic Manafolds
Next we proceed to the definition of Kahler structure on a complex manifold X with J as above. First of all we assume given a Riemannian metric g on X such that if we restrict g to any T p thern J preserve g (or J is orthogonal with respect to g), meaning g(Jv,Jw)= g(v, w) for all v,w E T p .
The standard example is Cnwith the previous J and
&,. .
*
, ayn a being
orthonormal with respect to g. Next we want to construct a 2-form w on X using g and J : we define w(v, w)= -g(v, Jw) for v,w E Tp.
Equivalently, w(v, Jw)= g(v,w ) . Then, w(w,v) = -w(v, w) because W ( U , v) =
-g(v, Jv) = -g(Jv, J 2 v )
= g(Jv,v) = g(v,J v ) = 0.
Note that if v # 0, g(v,v) > 0 is equivalent to w(v, Jv) > 0: this says that w restricted to the 2-dimensional space with ordered basis v, J v is an orientation. In C" with Euclidean g as before, w(&, &)= 1 (and w = 0 on other pairs) says that w = d x l A d y l . . * d x , A dy,.
+ +
Definition 1.4 A complex manifold ( X , J ) with Riemannian metric g such that J preserves g, is called a Kahler manifold (or has a "Kahler structure") if the 2-form w(v,w ) = -g(v, J w ) is a closed 2-form: dw = 0.
The simplest examples are Cn or open subsets U of Cn with Euclidean g , and quotients of Cn by a group of holomorphic isometries acting discon-
tinuously with no fixed points. Every Kahler manifold is locally "like Cn", where locally is to be interpreted in a suitable infinitesimal sense. For an explanation of this one may consult the book by Griffith and Harris, and we will just say that to prove certain identities involving d, J, d * , on a Kahler manifold it is only necessary to check them in the standard case of C". The identities we will need involve the following: Define dC= (J-' o d o J)/47r : AP -+ AP+l (SO d" = i(8- 8)/47r and dd" = ia8/2~).
Iterated Integrals, Chen's Flat Connection and x i
33
Note that dC as well as d takes real valued forms to real valued forms. Define (d")* = adjoint of d"(re1ative to ( a ,p ) =~ a A *P) so
,s
(dC)*= J-'d*J/4n
(as J* = J-').
The identities referred to above are: dd" = -d"d, dd"* = -dc*d, d"d* = -d*d"
+
+-
(however dd* d*d = A # 0, d"d"* d"*d" = A/167? # 0). J is defined to act on forms so that J(a A 0) = Ja A J p . Also
As an easy consequence of these identities, i.e., of the fact that d and d* each commutes (up to a - sign) with d" and dc*, we have a further decomposition of each Ap(X) into orthogonal subspaces: Ap(X) = 3Cp @ ddcAp-' @ ddc*AP@ d*dCAP@ d*d"*Ap+' (here dc* = J-ld* J/4n). Further, ddc maps d*dc*Ap+2isomorphically onto ddCAPc AP+' and is 0 on the other subspaces, and similarly each of the operators ddc*, d " P , d*dc*is an isomorphism on one of the above subspaces and is 0 on the others. This implies the important
Lemma 1.2 (d, d" lemma) Suppose Q E AP is d exact and d" closed, or else Q is dC exact and d closed. Then Q is dd" exact. Proof: We note that
Imd
= dd"Ap-2 @
ddc*AP
ker dC= 3Cp @ ddcAp-' @ d*dCAp, so Imd n ker dC= dd"Ap-'. Similarly Imd" n ker d = ddcAp-'. Another simple but important fact is: ker d n ker dC= 3Cp @ dd"Ap-' so that (kerd n kerdC)/ImddCis naturally isomorphic both to 3 C P and to H & R ( X ) .The isomorphism of this quotient vector space with H g R involves only the comples structure J of X and not the choice of a Kahler metric g (but assumes a Kahler metric exists).
34
Iterated Integrals and Cycles on Algebraic Manifolds
We can now complete the proof of Chen's theorem in the Kahler case by exhibiting the subspaces W , CP,Ep satisfying the condition l., 2 . , 3a., 3b needed. We begin by considering the subalgebra kerdC of AP for p = 1 , 2 , so kerdC = XP @ dcdAp-2 @ dCd*Ap. The subspace XP of Chen's theorem is taken to be just the harmonic pforms XP, the subspace C1 is defined to be dCd*AP,for p = 1, and the subspace E 2 is defined to be dcdAp-2, for p = 2, i.e., dcdAo. Thus d : C1 -+ E2 is an isomorphism. As noted previously, the conditions I.,2 . , 3a are valid in a Riemannian manifold, if we take X, C ,E to mean: harmonic, coexact , exact. Condition 3b was [(X1+C1)AC1]nkerdc E2.
To see this, we write
(X@ dcd*A) A dcd*A c dcA (since dc is a super-derivation and d C X= 0), and dcA = dcdA @ dcd*A, sodCAnkerd=dcdA=E.
Chapter 2
Iterated Integrals on Compact Riernann Surfaces
2.1
Introduction
Our next main purpose is to study Chen's connection 8 more closely in the lowest-dimensional case, namely when X is a compact Riemann surface with complex structure J and (Kahler) metric g (note that the Kahler condition dw = 0 is automatic here since X has real dimension 2). To do this we will make many simplifying assumptions on the terms of 8 on which we will concentrate, in particular we construct a simplified version 8 of 8; more precisely the series defining the homomorphism of 7r1 ( X ,xo) given by 8 will contain some of the terms of the series given by 8. A main reference will be [Harris, 1983~1.
2.2
Generalities on Riemann surfaces and iterated integrals
Note that the * operator on 1-forms of X is conformally invariant, so depends only on the complex structure and not on the choice of metric in its conformal equivalence class (this is always true in the middle dimension), in fact * = J-' on 1-forms here. Thus,
X1 = k e r d n k e r d * = kerdnkerd', C1= d"d*A' = d'dA' = $A2 = d"Ao, E 2 = dC' = dd"Ao, so all these subspaces are definable by d,d" alone, when X is a Riemann surface. Now let a1,a2 E X1 satisfy
35
36
Iterated Integrals and Cycles on Algebraic Manafolds
equivalently,
is exact, so
a1 A a2
01
A
a2
+ dQ12 = 0
for a unique a12 E C1 = dcAo. Let now A be the associative algebra with 2 generators x,y and relations x2 = y2 = xyx = yxy = 0.
A has R-basis l , x , y,xy,yx and contains the Lie algebra L = basis x,y, [x,y] (L=free Lie algebra on generators x,y). Let
with
e be the L
valued 1-form
Then 1 - -[eye] =o 2
de+ and so
e defines a homomorphism
t : 7rl(X,xo)-+ G = exp(L) = elements of the form exp(ax
+ by + c[z, y]) in A .
t is in fact a homomorphism of the fundamental groupoid (not necessarily closed paths, modulo end-point preserving homotopy) into G. This implies (1) (2)
J7172ai = J,, ai + J
s,
-
-
(3)
J,-l
72
ai,
+ Q12
(Q19 Q Z )
J,, (a1,a2) + a 1 2 + J
72
(a17 a2)
+ Q12 + J,, a1
JT2
a21
+ a12 = - J,(a1, a?,)+ a 1 2 + J, a1 J, 0 2 .
(a1,m)
Using (2), (3) we derive the following formula for changing base point in 7rl: let 20,xi be base points, 1 a path x1 to 20 (up to homotopy) y E 7r1(X,xo), 1yZ-l E 7rl(X,X I ) , then
Our aim is to simplify the algebraic situation by making further asai € Z for all sumptions on a1 , QZ, y. First, we assume a1,a2 satisfy: y E 7r1(X,xo). We denote this by ai E 3C:. We retain the assumption
&
37
Iterated Integrals on Compact Riemann Surfaces
Jx a1 A a2 = 0 and introduce a subgroup the kernel of
Thus
(Xi8 Xi)’of Xi €3 Xi,namely
and as before,
satisfies
Such a form a12 is defined for any element of that if we write
I ( % , a2;Y) =
I
(01, a2)
+ a12,
(Xi8 Xi)’.
mod
Z,
in
We now note
R/Z,
then by (2), I defines a function of 2 variables
I : (Xi€3 Xi)’x
7r1 (
X ,zo)
+
R/Z
which is “bi-multiplicative”, and since R/Z is an abelian group, factors through a homomorphism still denoted I , or I z o ,
IZ0 : (Xi€3 Xi)’€3 7r1 ( X ,z o y b
+
R/Z.
This homomorphism depends on the base point 5 0 , as shown by formula (4). In order to remove the dependence on 20, we restrict IZo to a smaller subgroup: namely the kernel of the homomorphism
(4) then shows that the definition of I on the subgroup is unchanged if we replace each yi by 17il-l. We can then replace 7r1(X,20)by H i ( X ;Z) and use the Poincar6 duality isomorphism H1 ( X ;Z)+ Xi (which we define so that [r]H a y = a3 implies
for all a E XI). Finally we denote the domain of I (the kernel just defined) as (Xi8 Xi €3 Xi)‘.
Iterated Integrals and Cycles o n Algebraic Manifolds
38
This group is the kernel of
Xi €3 Xi c3 Xi -+ Xi @ Xi @ Xi
We note that if a ~az, , a3 E satisfy then a1 @ 0 2 €3 a3 E (!Xi 8 Xi 8 Xi)’and
I(a1 @ a2 €3 0 3 ) =
I
5’
( a ~a2) ,
+
a( A
a12,
aj
= 0 for i , j = 1,2,3
mod Z
(where y E 7r1 (X, ZO) corresponds to ag). Next we would like to replace the domain of I by a subgroup (A3%;)’ of the third exterior power A3(Xi),defined as the kernel of a homomorphism to Thus consider the commutative diagram
x;.
0
- - 1@3 (XZ )
I
(Xp3
X;@x;@X;
-
0
Here
and j3(a @ p @ y) = cy
+ /3 + y. j z is the usual homomorphism and induces
31.
We will show that 21 can be defined as a homomorphism (A3%;)’
+
R/Z such that for triples al,az,ayg E Xi satisfying the special condition that for i = 1,2,3, ai is Poincar6 dual to a simple closed curve Ci,and C1, Cz, C3 are mutually disjoint, then
Lemma 2.1 Let Ci, i = 1 , 2 , 3 be disjoint oriented simple closed curves o n X . Then their harmonic Poincart! dual forms ai satisfy ai €3 aj @ (Yk E (Xi@’)’ and I ( a l @a2 @ a3) is invariant under cyclic permutations and is 0 (in R/Z) i f any two of the a’s are equal. Proof: Consider the surface Xi with two boundary components Ci,C: obtained by cutting X along Ci. More precisely, the boundary d X i is the
39
Iterated Integrals on Compact Riemann Surfaces
disjoint union of C,! and -C!, and to a point p on C, correspond two points p’ on Ci, p” on Cr.
Now integration of a( on X i from a base point 50 to a variable point 5 defines a single-valued (harmonic) function hi such that u ) hi@”)- hi($) = 1, and b) dhi = ai, (the result of integration is independent of path 20 to 5 on
Xi). Let 7 be any 1-form on X (continuous). By u ) we have c)
Let 77 = - a j , k , which is coexact and so orthogonal to the harmonic form *a,; then a
,s,
’
Rearranging terms, we get
+
Iterated Integrals and Cycles o n Algebraic Manifolds
40
which is cyclic invariance. Also
is clear, so we have skew symmetry of I(ai 8 aj 8 C Y ~ under ) all transpositions, but furthermore we have vanishing of this iterated integral if C Y ~= CYk since
Thus we have vanishing also if any two of the proof. 0
CY
are equal, concluding the
Lemma 2.2 Consider triples of disjoint simple closed curves C1,C2, C3 as above and write P = A3(Xi)',C1 A C 2 A C 3 E P instead of C Y ~ A C Y ~ EA P . Such "decomposable elements" generate A3(3Ci)' = P . Proof: Let Al, B1, . + ., A g ,B, be standard simple closed curves in X , where we think of Ai as going around the i-th hole, and Bi going around the corresponding i-th handle with intersection number Ai . Bi = 1 and all other pairs being disjoint. Consider three types of elements Ci A cj A a) 1 5 i < j < k 5 g and each C1 = Al or Bl. It is clear that such elements are as described above, belong to P , are linearly independent over Z,and form a basis of a direct summand of A3(Xi),and so a direct summand of P . There are 2";) of them. b) (Ai A l ) A (Bi - B1)A c k where i, k are distinct and > 1, and c k = Ak or Bk. The curves corresponding to Ai +A1, Bi - B1 can be taken as, respectively, surrounding both the first and i-th hole, or going around a handle separating these two holes (imagined as next to one another). There are 2(g - l ) ( g - 2) such elements. c) (Ai A2) A (Bi - B2) A C1. Here C1 = A1 or B1 and i > 2. These are of the same type as b) and there are 2(g - 2) of them. Modulo type a ) elements we may replace a type b ) , (Ai A1) A (Bi B1) A c k by (Ai A Bi - A1 A B1) A c k and similarly replace type c) by (Ai A Bi - A2 A B2) A C1 (where i > 2). These new elements just defined are still in P and belong to the subspace of A3 spanned by decomposables Ci A C j A C k where each Cl is A1 or B1 and two of the indices i,j , k are equal. Thus they belong to a direct summand complementary to that of the type a) elements, which involve 3 distinct
cr,:
+
+
+
Iterated Integrals on Compact Riemann Surfaces
41
indices. They are easily seen to be linearly independent and span a further direct summand. Finally, the total number of elemnets is (”) - 29 which is the dimension of P , concluding the proof of this lemma. 0 Corollary 2.1 Let D be the subgroup of [ ( X i ) B 3 ) ] generated ’ by all elements of type corresponding to C1 @ C2 @ C3 where the Ci are disjoint simple closed curves in X . Then jl :
1 ~ I3 (Xz ) + A3(X$)’= P
mups D onto P and 2 1 (twice the iterated integral homomorphism I : (XiB3)’+ R/Z) when restricted to D factors through jl : D 4 P and defines a homomorphism denoted
v =21:P
+ R/Z.
Proof: Clearly D is invariant under the permutation group action on ( X i ) @ 3and on ( ( X i ) @ 3 ) ’ .We have shown that for any d E D and u any transposition, I(d a(d))= 0. The kernel of j 3 : D + P is the intersection D n K where K is the subgroup of (Xk)B3generated by all “monomials” m = x @ y @ z with two of the factors z, y, z equal. Let S be the subgroup of ( X i ) @ 3generated by all elements of the form t ~ ( twhere ) u is a transposition. Then 2K c S. If d E D n K , d = k E K then 2d = 2k E S and
+
+
21(d) = 1(2d)= 0 since 2d
E S nD
and I = 0 on S n D. 0
We can now look a t the situation with Chen’s connection 8 and its homomorphic image and compare it with the usual Abel-Jacobi map Recall that induced a homomorphism
e
s
11 : .rrl(X,50) -+ G = exp(g)
where G is a simply-connected 3-dimensional nilpotent Lie group (over JR and g is a Lie algebra with generators z,y, with [z,g] # 0 but all further commutators 0.
e = a1 @ z + a z@y+a128 [z,y] (a1 A a:!
+ do12 = 0 ) .
42
Iterated Integrals and Cycles on Algebraic Manifolds
For y E .rrl(X,ZO) such that
JT
ai = 0, i = 1 , 2 we have in G = exp(g):
The usual Abel-Jacobi map is just the “first stage” of Chen’s construction, however we will apply it t o a different connection
with values in a 3 dimensional commutative Lie algebra c H I ( X ;R),hi E HI’ Shj ai = bij, where a( E X i ( X ) and a1 A a2 A a3 E P . We then obtain a map, the Abel-Jacobi map A : X 4 R3/Z3 if we choose a base point zo E X , namely
A(x)=
qx
a17 / x
50
10
02,/z a31 xo
mod Z3.
We write R3/Z3 for the Lie group exp(g) (g= Abelian Lie algebra spanned by homology classes C1, C2, C3 Poincar6 dual to the a i ) modulo the image of T I . Using this map A , we see that if we pull back to X , by A*, the standard 1-forms ti = dxi on R3/Z3 we get A*(dxi)= ai. Thus if we assume Sxai A a j = 0, i , j = 1 , 2 , 3 , then the cycle A ( X ) c R3/Z3 is homologous to 0 in R3/Z3:
A ( X )= d D for some(non-unique) 3-chain D in R3/Z3
2.3 Harmonic volumes and iterated integrals We can now compare the two maps X --+ Gllattice using R3/Z3 as follows:
Definition 2.1
e and A : X
-i
The (co)volume or “harmonic volume” corresponding to a triple a1,a2, a3 E X i ( X ) satisfying cri A aj = 0 for i # j is the “volume’’ mod Z of any 3-chain D in T3 = R3/Z3 satisfying d D = A ( X )
,s
43
Iterated I n t e g m k on Compact R i e m a n n Surfaces
as 2-chains] i.e.,
where A*(I =) ai and c1 A <2 A <3, regarded as cohomology class, generates H 3 ( T 3 ;Z). We orient T 3 using A Ez A 6 , = dxi on R/Z.
Theorem 2.1 With the above notation, and assuming the ai Poincare‘ dual to disjoint simple closed curves Ci as before,
so harmonic volume equals iterated integral. Proof: Cut X along C3 as before, obtaining X 3 with boundary C; - Cg. Let h3 be the R-valued function on X 3 defined by: h3(x) = “0 123. Recall that
s“
For i = 1 , 2 we define hi : X -+ R/Z again by the formula hi(.) So hi are also defined on X3. Denote
FL : xs
-+
s,:
ai.
R2/Z2 x R = T2 x R
h‘ on C;
the map given by (hl , ha , h3). So &I”) =
.
=
h’(p’)
Cg are related by
+ (O,O11)
restricted to Ci is null-homologous since
Since
7r1
and H1 for
T2x R are isomorphic h’ extends from C; -+ T2 x R
to a 2-disk B‘ with boundary CA:
extended
: X3 U B’
Since we have a homeomorphism C i a copty B” of B’ by:
jE(b”) = h’(b’)
-+
-+
T2 x R.
C; we can extend
+ (O,O, 1)
from C i to
44
Iterated Integrals and Cycles on Algebraic Manifolds
for corresponding points b‘, b“ in B’, B”. We have now filled in the two in X3 with two disks B‘ , B” and extended the map h‘ to the holes ci, filled-in surface X * = X ; = X3 uB’ U B” where 8x3 = Ci - C:, dB’ = C;,
c:
dB” = C:, as cycle X * = X i = X 3 - B’
+ B”.
We claim now that z ( X * ) bounds in T2x R: we just have to check that
Jx. i*(& A 52) = 0. But,
,s
The first term equals 01 A a2 = 0. The last two terms add up to (O,O, 1) and zero because L(B”) = ‘(El’) A & is unchanged by this traslation. Now consider the natural covering map
+
p : T2x R -+ T2 x (R/Z) = T 3
+
P*(‘(X*)) = p * ( i ( X 3 ) )- p*G(B’) p,L(B”) = p*x(X3)
(again the last two terms add up to 0 as chains).
so p , i ( ~ * )=) p,i(x3)= A ( X ) in T~ (we just use all the hi as W/Z valued functions). Further, as Z ( X * ) = boundary of a 3-chain D* in T2 x R, p * c ( X * )= A ( X ) bounds D = p ( D * ) in T 3 and the harmonic volume L E 1 A (2 A 53 =
where we denote p*& = dx3, monic volume is
23
L.
51 A 52 Ad23
= coordinate on R. By Stokes, the har-
(integration in T2 x R). Since @B”) is just K(B’) with 5 3 replaced by 1, (and are unchanged by this translation), the last two terms
23
+
Iterated Integrals on Compact Riemann Surfaces
add to
1
iE(B“)
ti A 1 2 =
45
L,,
p(h)A 6 * ( 5 2 ) .
Since B” is a disk, i*(&) is exact on it; on the boundary Ci, 6*(&)= a1 and so a1 extends to an exact form, say d f l on B”. Then
the iterated integral. Note that f l is only unique up to adding a constant, but Jcj’a 2 = Jc, a 2 = 0. Finally, the last two terms add to scs(a1,a2). The first term equals haal A LYZ, which we have shown to equal a 1 , 2 . So the harmonic volume equals the iterated integral.
sx, 2.4
sc,
Use of the Jacobian
Recall that (see for instance [Griffiths and Harris]) the Jacobian manifold J a c ( X ) can be defined as the compact torus H1 ( X ;R ) / H 1( X ;Z)and the Abel-Jacobi map A : X --+ J a c ( X ) , taking a base point zo E X to the identity element, is just the iterated integral map I I using only the Chen connection modulo commutators of g, i e . , using the commutative Lie algebra H 1 ( X ; R ) and the connection Etzl ai@J hi where {ai}is a basis of R i ( X ) and hi the vector space dual basis of H , ( X ; Z). Let X - denote the image of A ( X ) (denoted also by X ) , under the map g -+ 9-l on the torus J a c ( X ) . It is immediate that this map g --+ 9-l takes any translation invariant 1-form on J a c ( X ) to its negative, and so is the identity on translation invariant two forms, which are the harmonic forms in a translation-invariant metric. Thus both X(2.e. A ( X ) ) and X are cycles representing the same 2-dimensional homology class on J a c ( X ) and so their difference is a bounding 2-cycle: X - X - = d E , E a 3-chain (unique only modulo integral 3-cycles). The concept of the Jacobian variety J a c ( X ) can be extended to involve higher dimensional forms and cycles on a Riemannian manifold, for instance Y = J a c ( X ) . Given the 2-cycle Z = X - X - which is homologous to 0, we choose as before a 3-chain E such that Z = d E and consider all harmonic
Iterated Integrals and Cycles on Algebraic Manifolds
46
3-forms 0 on J a c ( X ) . The choice of E now defines a linear function
However E is only well-defined modulo integral 3-cycles1so that associated with 2 we obtain a homomorphism
v ( 2 ) : X 3 ( J a c ( X ) )-+ R/Z. v ( 2 ) is said to be a point on this intermediate Jacobian (consisting of all homomorphisms to R/Z). We will describe the effect of v ( 2 ) on a harmonic(=translation invariant) 3-form 0 in H 3 ( J a c ( X ) ;Z), which is isomorphic to h 3 H 1 ( X ;Z), only in the case that ,B is in the subgroup h 3 H 1 ( X ;Z)' = P , considered above. However P is in a sense the most important part of H 3 ( J a c ( X ) ;Z):
Proposition 2.1 (see [Harris 19831) A s subgroup of H 3 ( J a c ( X ) ; Z ) ,P is the primitive subgroup in the sense of Lefschetz, that is it equals the kernel of the homomorphism
H3(Jac(X)Z ; )4 H 2 g - ' ( J a c ( X ) ; Z) given by multiplication by ~
where w is the Kahler f o r m on J a c ( X ) .
g - ~ ,
Our main result is now
Theorem 2.2 Let X be a compact connected Riemann surface of genus 2 3, J a c ( X ) its Jacobian, v ( X - X - ) the point on the intermediate Jacobian of J a c ( X ) corresponding to primitive harmonic 3-forms. Let X - X - = dE on J a c ( X ) and let a1,a2,a3 E X i ( X ) be harmonic forms Poincare' dual to disjoint simple closed curves CI,C2,C3 on X . Regarding the ai also as l-forms on J a c ( X ) , we have
v ( X - X - ) ( a l A a2 A a3) =
s,
a1
A a2 A a3
(alz=unique coexact 1-form on X such that
a1
A a2
mod
Z
+ da12 = 0).
Proof: We use J a c ( X ) 3 T3, the natural projection (dual to the inclusion Z a l Za2 Za3 c Xi(J a c ( X ) ) .Note that T commutes with the (-1)map, g -+ 9-l. For the 3-chain E in J a c ( X ) , r ( E ) satisfies
+
+
a.(~) = .(XI
-
.(x-) = a(o- D-)
47
Iterated Integrals on Compact Riemann Surfaces
where Thus
D- =image of D under (-1) map of T3.See Definition 2.1 for D. r
A <2 A J 3 =
El
A
J2
A
J3
t i A E2 A J3
-
mod Z
ID-
JD
and
L-
Ac.3 = -
s,
G Ac.2 Ac.3
since the (-1) map sends J1 A J2 A J3 to its negative. This concludes the proof. In summary we have assigned to a compact complex Riemann surface X a point u ( X ) on the intermediate Jacobian of J a c ( X ) given by integration of primitive harmonic 3-forms on J ( X ) = J a c ( X ) over any 3-chain in J ( X ) whose boundary is X - X - , and have shown that u ( X ) = 2 x harmonic volume = 2 x iterated integral. However we have not yet shown that these quantities can be non-zero, (for g 2 3) and so we will prove a variational formula for them (their derivatives with respect to moduli), which will in particular prove they do not vanish identically. 2.5
Variational formula for harmonic volume
We will work over Torelli space , defined as the set of equivalence classes of pairs (compact Riemann surface X , symplectic basis for H1 ( X ;Z) relative to intersection number) where equivalence is given by complex analytic homeomorphism taking the given symplectic bases into one another. We will define a neighborhood of a pair by a type of deformation of the complex structure called a “Schiffervariation”. Torelli space is a quotient space of Teichmuller space by the action of a discrete group of complex analytic transformations acting freely, and is a complex manifold of complex dimension 3g - 3. A Schiffer variation of complex structure(described in [Schiffer and Spencer], Chapter 8, $1) is described as follows: fix a point t E X and a local coordinate z in a neighborhood o f t with ~ ( t=)0. Let z* be a local coordinate in a region of X which overlaps the domain of z in an annulus but excludes the disk IzI 5 p. X is obtained by identifying the domains of z and z* on their overlap by the identity map z* = z , while a new surface X * is obtained by the new
48
Iterated Integrals and Cycles on Algebraic Manzfolds
identification
z* = z
+ (s/z)
where s = e2ibp2 is any sufficiently small complex number. X corresponds to s = 0. A vector field in the neighborhood of the point t (i.e, of z = 0) is given by (l/z)d/dz(which is also expressible as d/ds at s = 0). This (meromorphic) vector field can be regarded as an element of H 1 ( X , K - l ) , K=canonical bundle (i.e., cotangent bundle) of X . The vector space H 1 ( X , K - ’ ) is dual to the vector space H o ( X ,K 2 ) of holomorphic quadratic differentials q(z)(dz)2 on X , the pairing being given by the residue at t: 1
= Restq(z )(dz) ( 1d/dz) ( q ( z )(dz) -d/dz) z z = Restq(z);dz = q(t).
The variation of complex structure just described is not the most general one, so we generalize it by first choosing on X 3g - 3 general distinct points ti, i = 1,.. . ,39 - 3; this means that we choose the ti so that any quadratic differential vanishing a t all of them must be zero. We will prove a little later that such (distinct) points t l , . . , tsg--3 exist. Then we choose sufficiently small disks around the ti so that they are disjoint, and use gluing maps
This gives us a 39 - 3 dimensional family of deformations parametrized by small s = (s1, . ,~ 3 ~ - 3 )The . Kodaira-Spencer theory (Kodaira-Morrow, “Complex Manifolds”, Chapter 2, Section 3) tells us that we obtain in this way all deformations sufficiently near X , i.e., a neighborhood in Torelli (or Teichmuller) space, whose tangent space at the point given by X is H 1 ( X ;K - l ) = dual space to H o ( X ;K 2 ) . We can now regard the Abel-Jacobi point v(X,) as a function of s with values in the intermediate Jacobian given by primitive 3-forms P ( X , ) . If we give these intermediate Jacobians the right complex structure (following Griffiths), v becomes a holomorpic section of a complex analytic family of complex tori. To compute the variation of harmonic volume with s, we use formulas of Schiffer and Spencer to see how a set of harmonic forms ~ 1 Q Z, , a3 vary with s (see [Harris 1983a], Sect. 5). Let the map X -+ (R/Z)3 used to
Iterated Integrals o n Compact Riemann Surfaces
49
define harmonic volume be denoted
(so a i = d h i ) . Let X *be the “nearby” deformed surface with corresponding
i* which is close to i. We need to calculate to first order in s the volume between the surfaces x * ( X ) and i ( X ) in T3 = (R/Z)3. This volume is a sum of volumes of small parallelepipdes with bases d i x d i in z ( X ) and (slanted) heights 6* Here d x x d 6 denotes the crossproduct of 3-dimensional vectors(2.e. the Lie bracket in the Lie algebra of SO(3) )combined with the wedge product of 1-forms. The volume element is thus the dot product
-x.
1 -(x* 2
-
g ) . ( d g x d;).
The formulas of Schiffer and Spencer lead us to the following calculation: as before, dhi = ai and there is a unique coexact form aij so that
Let
+ i * aij = ~ i j ( zZ)dz ,
aij
be the (non-holomorphic) 1-form of type (1, 0) with real part similarly let
aij,
and
be the holomorphic 1-form(i.e., of type (1, 0)). Consider the quadratic differential obtained by multiplying these (symmetric product) for distinct i, j , k: (aij
+ i * aij)(ak + i * a k ) = a i j b k ( d z ) 2 .
This is a non-holomorphic quadratic differential q i j k ( z ) ( d z ) 2= Q i j k . We write q i j k = Q i j k / ( d ~ ) ~ . However, minus the sum of these over the 3 cyclic permutations of ( i , j ,k) = ( 1 , 2 , 3 ) ,i.e., Q ( a l , o z , a g ) defined as -
(a12 (1,231
+ i * (Y12)((Y3+ i * a 3 )
Iterated Integrals and Cycles on Algebraic Manifolds
50
is holomorphic, and Q defines an R-linear map on h 3 X i ( X ) ' @ z R= P g Z R : Q :P @ z R + H o ( X ; K 2 )
(where P c X g ( J ) as before). This R-linear map is C-linear if we choose the Griffiths complex structures on P @zR (see [Harris, 1983~1,Sect.4) which is defined as follows on the larger space A3X1( J ;R) = A3X1( X ;R): first, define a complex structure j l on X1(X) as j l = -*, so j 1 @ 1 on holomorphic (1, 0) forms is multiplication by i. Then on A 3 X 1 ( X R) ; let j 3 be
+
I .
j 3 ( A A B A C ) = - [ji( A ) A jl (B) A jl ( C ) j l ( A ) A B A C 2
+ A A j l ( B )A C + A A B A j , ( C ) ] .
On (H39O @ H 2 i 1 ) ( J )this is multiplication by i, and on H'i2 EIHoy3 it is multiplication by -i. We then have the following theorem (loc. cit. Theorem 5.8)
Theorem 2.3 volume
for s
Let X =
= ( ~ 1 , .. . , ssg--3)
a1 A a2 A a3 E
P and consider the harmonic
near s = 0 . Then
(Here t j are the 39-3 general points on X , with z j local coordinates centered at tj and the function obtained by dividing the quadratic differnetials by ( d z j ) 2 into which we substitute t j ) . I(X,s) is a holomorphic function of s for each A. We can explicitly calculate the differential 6 1 : P 8 R = A 3 X k ( X ) ' + H o ( X ;K 2 ) of I at s = 0 on the locus in Torelli space of hyperelliptic Riemann surfaces
51
Iterated Integrals on Compact Riemann Surfaces
if the branch points e l , . . . , en (distinct complex numbers) satisfy n
i=l
is a symmetric function of Argz, and prove that in this case
6 1 : A3Xk(X)’--+ H’(X; K 2 ) has as image the whole -1 eigenspace of the hyperelliptic involution, i.e., this image is as large as possible [Harris 1983~1. To summarize, if X is hyperelliptic then v = 2 1 vanishes on X - X(since X- = X) and so I takes on values 0 or mod Z,and is therefore constant as R/Z valued function on the (connected) hyperelliptic locus, whose complex dimension is 29 - 1. The space of tangent vectors to Torelli space a t (X) which are “normal” to the hyperelliptic locus is 9-2 dimensional and dual to the (-1)eigenspace of the hyperelliptic involution of X acting on the quadratic differentials of X . Regarding I = I , as function of the moduli parameters s (and constant on the hyperelliptic locus) and regarding its differential S I as linear mapping on the (-1)eigenspace, 61 is injective on this normal space, if X varies in a non-empty open subset of the hyperelliptic locus. More precisely, Torelli space is a complex manifold and over it we have the family of complex tori Horn(P,R/Z) which is a complex vector bundle modulo a locally constant family of lattices (certain integral coefficient homology groups). Iterated integrals define a holomorphic section of the vector bundle whose differential is injective in the normal direction to an open subset of a submanifold (along which the section vanishes). In particular the section of the bundle of complex tori is not identically zero. We still have to prove the existence of divisors D = tl . . . t3g-3, with distinct ti, such that any holomorphic quadratic differential vanishing at every ti must be 0 , in other words
i
+
HO (K 2 D - 1 ) 49
=
+
(0).
The degrees of these divisors are as follows: deg K = 29 - 2 , deg K 2 = 4, degK2D-‘ = g - 1. By Riemann-Roch, the dimensions satisfy
-
hO(K2D-l) - ho(DK-1) = g - 1 - ( 9 - 1) = 0. Thus we must find D so that h’(l3K-l) = 0: this means that in the linear equivalence class of DK-’ (which has degree g - 1) there is no effective
52
Iterated Integrals and Cycles o n Algebraic Manifolds
divisor D’. Equivalently, D is not linearly equivalent to KD’ for any effective divisor D’ (of degree g - 1). Consider now the (Abel-Jacobi) map u from divisors of degree 39 - 3 to the Picard variety P Z C ~of~isomorphism -~ classes of line bundles of degree 39 - 3 (isomorphic to J a c ( X ) ) :
u : ~ y m ~ g --+~ ~( i~c )~ g - ~ . All fibers of u are projective spaces P29-3. Within we have the subvariety given by all line bundles KD’ (degD’ = g - l), which is u ( K Symg-’(X)) and so has dimension 5 g - 1. Thus we just need to choose D in the open subset of S ? ~ r n ~ 9 - ~such ( X ) that u ( D ) E - u(K Symg-l) and none of the 3g - 3 points are equal. So we have proved the existence of the “general” divisors D , and the nontriviality of the section I of the family of primitive intermediate Jacobians, taking a point corresponding to X to the Abel-Jacobi image v ( X ) . However this result does not directly exhibit a Riemann surface X where this AbelJacobi image v(X) is non-zero. A little further on in this chapter we will exhibit such an X , namely a well-known algebraic curve (of genus 3)
x4
+ y4 = 1
defined over Z: to do this we will just calculate some iterated integrals. In the next section we recall the connection (due to Hodge and Weil) of the Abel-Jacobi map and equivalence relations on algebraic cycles, then give the application of our methods to this problem.
2.6
Algebraic equivalence and homological equivalence of algebraic cycles
We want to look at the problem of whether a complex algebraic 1-cycle 2 i.e., of complex dimension 1 on a smooth projective complex algebraic variety V is algebraically equivalent to 0 if it is known to be homologous to 0. We will only give a rather vague, intuitive idea of what algebraic equivalence to 0 means - for more details, see Fulton’s “Intersection Theory”. We will use the following definition: suppose the complex 1-cycle 2 (considered as topological 2-cycle) is the boundary of a topological 3-chain W (i.e., W has topological dimension 3). We then say that 2 is homologous to 0. Suppose further there is an algebraic (or complex analytic) subset S of V of complex dimension 2 such that the topological 3-chain W lies on S: we then say that 2 is algebraically equivalent t o 0 (in V). A more
Iterated Integrals o n Compact Riemann Surfaces
53
intuitively understandable situation is the following: suppose Zo,Z1 are two irreducible one-dimensional complex subvarieties of V which are two members of a family of subvarieties Zt, t E T , all Zt being l-dimensional (over C ) and T being a connected algebraic curve over @. In this latter case the union, in V , of the Zt will be the subset S of complex dimension 2. (If further the parameter space T is a rational curve then 20, Z! are said to be rationally equivalent.) Here we let Z = 21 - 20. Hodge and Weil (see Weil, Collected Papers, Paper 1952e and comments on it) proposed the following criterion for algebraic equaivalence to 0 of the complex l-cycle 2 on V - the criterion is clearly necessary but it is not obviously sufficient: if Z = dW on S then for every complex (3, 0 ) form w on V , the restriction of w to S is clearly 0 (since S has complex dimension 2) and so &,w = 0. However since the chain W is not unique but is only unique modulo topological 3-cycles, the criterion is that if Z is algebraically equivalent to 0 then the linear function it defines on complex harmonic forms of type (3,O) is 0 modulo the linear functions of integration over all topological 3-cycles. Denoting real harmonic primitive 3-forms by P 3 , we form P 3 8~C and use the Hodge decomposition p3
=~
3 3@ 0 ~271 ~ 1 7@ 2 ~023.
In fact P3l0= X3>O, = R?>3because w E X'*'. For example, if V = Jac(X), X a genus 3 curve, then X1io and Xoil each have dimension 3 over C, X3l0 and are each l-dimensional, and X2y1and X1l2are each 9-dimensional. P3 has dimension 14, P310 = X310 and Poi3 = X0>3are l-dimensional. The primitive intermediate Jacobian of X we use is (* denoting complex dual) (p3,O
P2,1)*
Image H3(Juc(X); Z) and we have an Abel-Jacobi type homomorphism
(P39* Complex l-cycles on Jac(X), homologous to 0 + Those algebraically equivalent to 0 Image H3(Juc(X);Z) * In general, the image of H3(Juc(X);Z) in the one dimensional complex vector space (P3l0)*(for X of genus 3) need not be discrete. However we would like to find examples of X such that this image is discrete: this will be the case if for a basis y k of Hl(X;Zand a @-basis 9, of f l ' ~ o ( X ) (holomorphic l-forms), the period matrix, with entries JTlrc 9 j , has entries
54
Iterated Integrals and Cycles on Algebraic Manifolds
in a discrete subring of C. Then on J ( X ) the (3,O) form 81 A O2 A O3 has integral over each 3-cycle a 3 x 3 minor (determinant) of the 3 x 6 period matrix and this is again in the discrete subring of C. We can then try to calculate the image of the 3-cycle X - X - in Image Hs(Juc(X);Z) and see if it is non-zero. This program can be carried ot for certain X with large automorphism group, for instance the Fermat quartic (P3l0)*/
x4 + y4 = 1 using formuIas for the period matrix entries in a paper of B. Gross and D. Rohrlich (the entries are given by the Gamma function). The details of the calculation will be given in the next section. Finally we find that
is non -zero in the group (P370)*
Image
H3
-
c Z(i)
and so X - X - is not algebraically equivalent to 0. However, further work by S. Bloch, [Bloch], using Z-adic cohomology, was needed to prove that X - X - has no (non-zero) integer multiple which is algebraically equivalent to 0. It had previously been shown by P.Griffiths using a variational method, that in certain families of cycles homologous to 0, all but some subset of measure 0 of this family of cycles were not algebraically equivalent to 0. However this was basically an existence result, and could not exhibit particular cycles, much less ones defined over Z or a number field. It was thus important to exhibit examples such as the Fermat quartic on which arithmetic conjectures of Bloch about the “Griffiths group” could be tested. Iterated integrals on this particular curve are further studied in [Harris, 19901 where the curve is instead described as the modular curve Xo(64) and its holomorphic l-forms are certain theta functions. Iterated integrals of a class of theta functions, those having harmonic polynomial coefficients, are there shown to be a more general class of functions: theta functions with rational function coefficients. These latter are not modular forms but their Mellin transforms satisfy a certain type of functional equation.
Iterated Integrals on Compact Riemann Surfaces
55
Calculations for the degree 4 Fermat Curve
2.7
This section will reproduce an unpublished paper giving the calculation announced in [Harris, 1983b]. The main result of this calculation is that v(01 A 0 2 A 0,) in the group C / Z ( i ) is (up to multiplication by -i), the real number ( mod Z) 1
21
((1- x4)-’/2dx, (1- x4)-3/4dx)/
1
1
(1 - x4)-1/2dx
1
1
(1 - x4)-3/4dx
where the numerator is an iterated integral. Since the numerator is an integral over a triangle 0 5 x 5 y 5 1 and the denominator is over the square 0 5 x 5 1, 0 5 y 5 1, it is easy to estimate that the real number above is not in Z Further work by S. Bloch, C. Schoen, and D. Zehinsky has been done to study this and related examples by various techniques of arithnetic geometry (see [Bloch] for an initial paper).
Homological versus algebraic equivalence in a Jacobian We will consider the problem of deciding whether two homologous algebraic p-cycles C,C’ in a compact non-singular algebraic variety V over C are algebraically equivalent (i.e., roughly speaking, the cycle C - C’ can be “continuously” (algebraically) deformed into the cycle 0; see [3]for a precise definition). A direct method of proving algebraic non-equivalence (for p > 0) is to consider a singular chain D of (topological) dimension 2p 1whose boundary is C - C’ and integrate holomorphic ( 2 p 1)-forms over D thus obtaining an element of the quotient group of the dual of the (2p+ 1)-forms by the subgroup of periods (integrals over ( 2 p 1)-singular cycles). If this element of the quotient group is non-zero then C, C’ are not algebraically equivalent (compare with Hodge’s letter of 1951 given in [S]). We carry this out on the example of the Fermat curve F : z4 y4 = 1 for C , its Jacobian J for V, and for C’ the image of C under the map of J given by the groupstructure inverse. The method for calculating integrals over the 3-chains D is that given in our paper [5] and amounts to calculating iterated integrals of holomorphic l-forms on the Riemann surface C. The special feature of the Fermat curve of degree 4 is that its normalized period matrix has entries in a quadratic imaginary field and so the periods of holomorphic 3-forms on J generate a discrete subgroup of C: consequently, the integrals in question need only be calculated to sufficient accuracy to see that they are not in
+
+
+
+
56
Iterated Integrals and Cycles on Algebraic Manifolds
this discrete group. Thus the same method can, in principle, be applied to any curve with normalized period matrix in a quadratic imaginary field. The above direct method has not been carried out before now, perhaps because of lack of a formula for the integrals involved. Instead, Griffiths([2]) developed another method, by which he gave the first examples of homologous non-algebraically equivalent cycles, namely consideration of families V, of varieties depending on parameter t , and differentiation with respect to these parameters. However, this method only allows one to prove that a generic member & of the family carries such cycles (generic usually meaning that t lies in the complement of a countable union of proper subvarieties). All subsequent examples of algebraic non-equivalence have followed the Griffiths method. In particular, Ceresa [l]recently proved algebraic non-euqivalence for a non-singular generic curve C and C' = inverse (C) in J ( C ) by this differential method; our paper [5] contains another way of doing this. The problem for curves C over Q was mentioned to us by Spencer Bloch; I wish to thank him, Ron Donagi and Gerry Washnitzer for conversations. 1. Let C be a non-singular curve of genus 2 3, and let HPiQ denote the @-space of harmonic (p,q) forms on J ( C ) . Let P+ denote the subspace of primitive forms in H3>0 so that H3l0 C P+, P; its complex dual, and L the lattice in P; of all linear functions obtained by integrating over integral 3-cycles in J . P;IL is a complex torus and the cycle C - C' determines a point v on this torus, namely v(q5) = q5 (mod "periods"), for 4 E P , where d D = C - C'. The embedding of C in J , and hence of C', depends on the choice of a base point po E C, but u is independent of base point because we deal only with primitive 3-forms. We recall from [5] how to calculate v in terms of iterated integrals in C of real harmonic 1-forms a , p: if a A p is an exact 2-form on C,a A p = d v where q is uniquely determined by being required to be orthogonal to all closed 1-forms on C, and if 1 is a path, parametrized by t E [0,1], then the iterated integral is
+
H2i1,
,s
I(&, p; 1 ) depends only on the homotopy class 1 (with fixed endpoints). If 1 .1' denotes a product path, 1-1 the path inverse to I , and c a closed path
Iterated Integrals o n Compact Riernann Surfaces
57
then
I ( a , P ;1 . 1 ' ) I ( a ,p; 1 . 1-1)
=
I(., P; 1)
+
I(Cy,P$)
+ S,
Q
. S,, P
(2.2)
=0
I ( a ,P; 1 * c .Ip1) = I ( a ,p; C)
(2.3)
+ S,
Q
*
J, P - S, p . J, Q
(2.4)
I ( Q ,P; I) + I ( P , a;1) = J a S, P
(2.5)
( a ,,O) may be replaced by any element of (Xc g X ~)'where X denotes real harmonic 1-forms and (YC @ X)'is the subspace of X cg YC annihilated by the real linear function J, Q A P. Let Xz denote real harmonic 1-forms with periods in Z and (X@z X)'the corresponding subspace of Xz @ Xz. Fix a base point p o E C and let ~ l ( C , p be~the ) ~abelianized ~ fundamental group, isomorphic to YCz under Poincar6 duality. Then (2.1), (2.2) allows us t o define a homomorphism
r : (Xz@ Xz)l@7r1(C,po)ab
+
R/Z
(2.6)
equivalently: (Xz @ YCz)' @ Xz R/Z, (1(a,,B;C)= J,[(Ja)P - 171 mod Z). Let Z ( i ) denote the Gaussian integers, 9tz(i) = X @ Z(i) the complexvalued harmonic 1-forms with periods in Z ( i ) ,and Hi;!) the Z ( i ) submodule of holomorphic 1-forms with periods in Z ( i ) . On tensoring (2.6) with Z ( i ) we get ---f
p
@Z(i)
(%(i)
%(Z)Y
@Z(i)
(m(C,po)ab @ Z(i))
: ( X Z ( i )@Z(i) XZ(i))/@ Z ( i ) %(i)
-+
-+
C/Z(i)
C/Z(i).
(2.7) (2.8)
In [ 5 ] , Sec. 2 , we prove the following: in the commutative diagram with exact rows 0
-
0-
(Xz@ xc,@ Xc,)'
1j l pz
where, denoting
sc x
-
Xz 63 Xz @ Xc,
--+
A
1 .i2 h3(Xc,)
B
Xz e3 Xz e3 Xz
1 .i3
P
Xz
+
-
0
-0
y by x . y for x,y E XZ,
P(x @ y @ 2) = .(
P(xA y A Z) = (.
*
y)z Y)Z
@
(y . 2)x
+ (9
Z)X
(z . x)y
+ (Z
*
X)Y
is the natural homomorphism and induces j l ) we have: jl is surjective, (3tz @ XZ63 Xc,)' is a subgroup of the domain of f, and 2 1 restricted to
j2
Iterated Integrals and Cycles o n Algebraic Manifolds
58
(Xz8 xz @ Xz)’ factors through jl: there is a unique homomorphism D :P
+
R/Z with
v oj1
= 2 1 and
a commutative diagram
(Xz EI Xz 8 Xz)’ c (Xz@ Xz )’@ xz % R/Z
/+
\
(2.9)
P and on extending scalars to Z ( i ) z 7 : &(Z)
(2.10)
+ C/Z(i)
(2.9), (2.10) can be rewritten as
Hom(Pz, Z) is the restriction to Pz of H 3 ( J ;Z) = Hom(H3(J;Z), Z) (since Pz is a direct summand of H 3 ( J ;Z)), and similarly, Homz(i)(Pz(i), Z ( i ) ) is the restri tion of H3(J;Z ( i ) ) . It is shown in [6] that the u of the beginning of this section is the same as the V of (2.9), (2.10). From now on, we will write only u or I for these homomorphisms. We will only be interested in the restriction of u of (2.10) to certain holomorphic 3-forms, namely (Hi&) c (Xz(i))’. We assume
@, and let 81, . . . , Og be a Z(i) basis from now on that H1yo= Hi;:) 8qi) for H&:). Then A A 6,) = 21(ei EI 8 0,) is calculated as follows: by Poincar6 duality, the dual of 6 , is x a r K , where K1,... , Kz9 is a Zbasis of H 1 ( C ; Z ) and a, E Z ( i ) . Let K , be the homology class of C, E T ~ ( C ; then ~ ~I ()&,8 0, EI 0,) = X U ,Jcr(ei,ej) ( mod Z ( i ) ) . u is determined only modulo ( h 3 ( H i & )Z, ( i ) ) , which is generated by the homomorphisms
e,
J
e,
s,,
oil A ei2A Bi3 = det[
Kpl A K P , A K p 3
Oil-] E ~ ( i ) .
For simplicity, let g = 3, then f is given by one complex number (a linear combination of iterated integrals) modulo Z ( i ) , and approximate evaluation of the integrals will suffice. It is clear that Z ( i ) could be replaced by an order R in an imaginary quadratic field, but it is essential that R be discrete in @ if approximate evalution of integrals is used.
2. Let C
= F(4) : x4
+ y4 = 1. [4]will be our main reference here.
Iterated Integrals on Compact Riemann Surfaces
59
We consider this Riemann surface as 4-sheeted covering of the x-plane brnached over x = ij, j = 0 , 1 , 2 , 3 , ( i = @ I 2 ) . Make cuts from ij radially out to 00, thus defining four sheets which will be labelled by the index S = 0 , 1 , 2 , 3 such that at X = 0, y = is on sheet S. Let a be the path going from 1 to i on sheet 1 and returning from i to 1 on sheet 0: a = (1i)l. (i1)o.
This is the path used in [4],and its homology class together with its transforms under certain automorphisms give a basis of H l ( F ; Z ) . The automorphisms are A , B:
A homology basis is
The intersection numbers are
K2j-1
0
K2j = 1, K,. o K , = 0 if
( T , s)
# ( 2 j - 1 , 2 j ) or (2j, 2 j - 1).
xr-l 9-4 Let qr,s,t Y dx. Then 771 = 7 7 1 , 1 , 2 , 772 = 7 7 1 , 2 , 1 , 773 = 772,1,1 are = ijr+ICsqr,s,t. Let holomorphic and a basis for H1io. Further AjBIC77T,,,t
(2.12)
where B(T,s) is the beta function
m.
Then (2.13)
and the other periods
J K p 77;
can be determined by using
Iterated Integrals and Cycles o n Algebraic Manifolds
60
We will take
Then, the periods of 0j over the cycles ()2.11 are
82=(1-i)(q;-q;) 133 = (1 -i)(qi - q ; )
0 0
0
-(l+i)
i
0
1 i
l+i l+i
-1 i-1
Using column operation, i.e., a change of basis of the K j over Z ( i ) , we get the new period matrix
The periods of 81 A82 A83 = $(2qi A 277; A2qi) over the last Z(i) basis of H3(J;Z(i)) clearly generate Z(i), and so 81 A 82 A 03 is a Z(i) generator of Hi$)(holomorphic 3-forms with periods in Z(i)). We will calculate v(81 A 82 A 83) = 21(01 A O2 A 0,) E C/Z(i). The Poincark dual homology 0 2 ( o =intersection number) class PD(82)is defined by PD(02)o Kj = so that
sKj
PD(82) = (1+ i)&
+ K3 - (1+ i)Ks - K5.
(2.16)
We will evaluate i(Ol A 133 A 82) = f(81 A 03 123I'D(&)) as the linear combination of iterated integrals (81, 03), where a, E ~1 (F;1) has homology class K j , with coefficients those in (2.16). It will be simpler to write the integrand (01,03) as (2q;, (1-i)(q: - 77;) = 2(1- i)(qT,$) - 2(1 -i)(v;, q;) and recall that 2 S,(q;, qi) = S , q; . q;. i) a3 = B2a has base point at 1,
saj
s,
Iterated Integrals on Compact Riemann Surfaces
61
i i ) K4 has the homology class of A - l B a but A - ~ B uhas base point at A - l ( l ) = -i. Let 1 = ( l i )be ~ a path on sheet 0 from 1 to i, then A-'(1)-' goes from 1 to -i and we may take a4 = A-'(1)-'
To calculate
. A-lBn . A-'(1)
$, note that a = B(1). 1-1
so that
iii) K5 = A2B2u,but A2B2ahas base point at A2(1) = -1, so a5 =
iv)
1 . A(1) . A 2 B 2 a .( 1 . A(Z))-l
62
Iterated Integrals and Cycles o n Algebraic Manifolds
+
/ / l.A(l)
rl;
17;
A*B-’a
1
8 -+(l+i)(-
-1 + i 4
Taking the linear combination given by (2.16),
+-14- i
+ -1+ -1- +- 3 i 2 4
~(0A 1 6’3 A 0 2 )
= -16i
i 2
J,($, 77;)
mod Z ( i ) .
To calculate sa(qr,17:) we will consider first the automorphism 3) of F : x4 y4 = 1.
+
Y
0 (X,Y)
=
(-,-) X&
1 ax
c7
l+i
(&= -)
Jz
((0l)l is the path on sheet 1 from x = 0 to x = 1.) g[(li)1]= (01)3(10)2 = B3[(Ol)o* B-’(Ol),’]
= aB(l),
(of order
Iterated Integrals on Compact Riemann Surfaces
(recall: 1 = ( l i ) ~Therefore, ).
Similarly,
Now
( using (2.2) which implies
Finally,
Recall:
63
Iterated Integrals and Cycles on Algebraic Manifolds
64
(where on sheet 0, (1- x 4 ) l l 2 = 1 when x = 0).
dx
1
" = B(1/4,1/4)
(1- x4)3/4
'[Ix
2
Jdh;'d*) = B(1/2,1/4)B(1/4,1/4)
dx
(1 -dt t 4 ) 1 / 2 ] (1- x4)3/4
r(94
Using: B(1/2,1/4)B(1/4,1/4) = -4,-
-
(3.6256099082)~
T J z
- _1
4'
and
X J z
dt 1 dx = 1.5113980 f .007, (1- t4)1/2 (1- x4)3/4 we get
l ( ~ $ , q ;= ) 2(.03886151) f .0036 - 1/4
v(ol A 03 A 0,) = 2i(el A 03 A 02) = -xi
I
(q;,7;)
= [1.2435683 f .00576] x (4) mod Z ( i ) .
3. Some further details on
First, the elliptic integral change of variable
J:
(0 5 x 5 1) trnasforms under the
2 2t2 sin 9 = - t =
1+P1
--dt
de
into the integral (where x
sin2e
+
1 cos28 ' 1
sinrp
= 1J-
1-t
4
=
+
d0
de
sin2 cp =
4cos2e (1 cos2 0 ) 2
1f z
Iterated Integrals on Compact Riemann Surfaces
(notation of Abramowitz and Steigun Bureau of Standards volume) (0 5 cp 5 7r/2 corresponds to 0 5 x 5 1). The iterated integral is then
(Since
&+ under x = JG&' sin 'p
Let 7 ) = 7r/2
-
becomes
cp then the integral is
To take care of the singularity of 1 + +3/2 1 m-vq 12
1
--
__
at
= 0, write
$J
+ -111,w
+
11,2 3!
-7/J4_ . .
160
61 7 ) w + .. . . 7 x 72 x 240
Since starting with sin 11, --I--+ 11, = (1
5!
+ Ul11, + a211,2 + . .
.)2,
we find
2.3!
sin 11,
2 ' 3!
+---
2.6!
160
5@ + . . 24*7!
61 11,6+ . . . 7 * 72 .240
65
66
Iterated Integrals and Cycles o n Algebraic Manafololds
and is five times differentiable. We write our integral as a sum of two terms:
and integrate by parts in the second integral, noting that the integral of the first factor vanishes at $ = 0, and the derivative of the second factor -1 while IF($ - $/;) vanishes a t $ = r / 2 . Thus the is ~ 1 - 1 / 2sinz(r/2-+) second integral is
and with the substitution $ = t2, we get
[2t2
1 t6 t10 ++ 7201 30 dl - 1/2 cos2(t2) dt.
Numerically (using 20 subdivisions and Simpson’s rule), we get 3.0186634/2 =1.5093 for this integral and for the first integral, get .0035718/2. Thus for s ( o l ) o ( r / 2 , r / l ) we get 1.5111 and for 2 B(1/2,1/4)B(1/4,1/4)
Jt[s:
dt (l-t4)1/21
(1-$)3/4
we get the
given before’
[l]G. Ceresa, “C is not algebraically equivalent to C - in its Jacobian, Annals of Math. vol. 117, 1983.
”
121 P. Griffiths, “On the periods of certain rational integrals 11, ” Annals of Math. vol. 90, 1969, pp. 496-541.
[3] P. Griffiths, “Some transcendental methods in the study of algebraic cycles,” Lecture Notes in Math., vol. 185, pp. 3-4, Springer-Verlag, Berlin, 1971. [4] B. Gross, D. Rohrlich, ”On the periods of abelian integrals and a formula of Chowla and Selberg, ” Inv. math. vol. 45, 1978, pp. 193-211.
Iterated Integrals on Compact Riemann Surfaces
67
[5] B. Harris,” ‘Harmonic Volumes,” Acta Mathematical vol. 150, 1983 pp. 91-123.
[6] A. Weil, Collected Works, vol. 2, pp. 533-534.
2.8
Currents and Hodge theory
In order to generalize to higher dimensions some of the calculations we have carried out on Riemann surfaces, we have to replace the technique of cutting up these 2-manifolds along curves and considering functions with jumps across the cuts by more general methods suited to higher dimensional manifolds and submanifolds. As preparation for doing this in the next chapter we review here some of the theory of currents and associated Hodge and de Rham theory. Let X be a compact C”” manifold of dimension n, oriented, without boundary. AP(X)=space of C“ p-forms (real valued).
Definition 2.2 A current T on X of dimension p (or degree n - p) is a n R-linear function T : A p ( X ) -+ R satisfying a continuity condition: there exists an integer k and a finite number of coordinate neighborhoods U j covering X such that if in U j we write a p-form 4 as 4 = 41(z)dz1 (I=multi-index) then for some constant C,
IT(4)l I c
cc I
SUP IdJ4il
lJl
( J again denotes multi indices, d’=partial derivative). We also say T is of order 5 k.
,s
4 A 7.
Example 2.1 Here k = 0.
Let
Example 2.2
S is an oriented Cm submanifold of X of dimension p , and
T
be a C““ n - p form on X and let T ( 4 )=
Here we write T = Ss = ‘
Iterated Integrals and Cycles o n Algebraic Manifolds
68
Further we can take several such S: say Sl,. . . , S, and real coefficients clr... c, and define T ( 4 ) = CZ, ci Jsi f;4. We write T = Ss. 1,
Example 2.3
&.
Let
X
= S2 = R2 U 00, g(z, y) = any
L1 function (say on
R2), e.g, 1% P = 2 and T ( 4 )=2,J g ( z , y ) 4 . Recall that T is said to have dimension p and degree n - p . We now define the exterior differential dT as the current of degree p 1 given by (for any p - 1 form p)
+
dT(p)= (-l)’T(dp). If we use the notation
,s
then ,6’ A dT is defined as ( - l ) P ,S dP A T in accordance with example 2.1 where T is given by an n - p form r. In example 2.2, T = 6s where S has boundary as,we get dT = (-1)’Sas. Next we look at dT for T given in example 3, where X = R2 U 00 = S2 and T is given by its value on a 2-form 4 on S2 as
Here dT = d log( &), as current of degree 1. If X is a complex manifold with complex structure operator J (which acts on tangent vectors and on differential forms), we define J ( T ) for a current T so as to agree with the definition of J ( I - ) for a form I- if T is given by r: since we have
( J ( 4A I-)= J ( 4 ) A J ( r ) ) ,we define
(4,J ( T ) )= (J-Y4)7T ) . As example we consider 9’ with polar coordinates r , 8 . Then
J ( d z / z ) = idz/z, J ( d r / T ) = -do.
Iterated Integrals on Compact Riemann Surfaces
69
If T is the current given by log(?), then J-'(dT) = 2dB. The operator d" is defined as d" = L J - ' 47f
SO
odo
J
=
(i/47r)(8-
a)
dd" = (i/27r)aa. If T = log? then d"logr2 = (1/27~)de dd" log r2 = d(dB/277) = 60 - 6,.
Thus, for a differentiable function 4(z, y) on S2,
We generalize the last example as follows: 1. On S2 = P1(C) we have the complex line bundle O(1) whose fiber over each 1-(complex) dimensional linear subspace C ( z 0 , z ~ of ) C2 is the dual space C(z0, z1)*. We have a hermitian metric on each fiber, given by: if 1 : C ( Z O , Z+ ~> C is linear then
Let now s be the holomorphic section of O(1) such that s(C(z0, zl))=linear function I , l ( z 0 , z l ) = 21. Then ( ( S I ( ~ is a function on P1 given by ( ( S I = (~ 1z112/(1z012 1t1I2) and: dd"logllsl12 = 60 - w where w is a 2-form on P1 w = 1. w is the invariant under the unitary group U(2) and satisfying (first) Chern form of O(1). Finally z1/zo is a meromorphic function on P1 with divisor div(zl/to) = (0) - (m). Write t = z ~ / z g :then we have an equation involving currents:
+
dd" log 1tI2= 6 d i ~ ( t l / t a ) . The following is a general fact, called the PoincarbLelong formula (see [Griffiths-Harris]): for any complex analytic line bundle L over a complex manifold X with hermitian metric on the fibers and s a non-zero section of L (holomorphic or meromorphic) with divisor div s = 2 = 20- Z, we have
70
Iterated Integrals and Cycles o n Algebraic Manifolds
where w is the Chern form defined so that it is a (1, 1) form representing the Poincar6 dual cohomology class to the homology class of div s. De Rham’s results for currents: The first result is that the cohomology of the space of currents for the operator d is the same as the cohomology of the subspace of differential forms: that is, the inclusion of A * ( X ) into the space of all currents (for compact oriented X)induces a cohomology isomorphism. In particular, if CY is a closed form on X and CY = d T for a current T then CY = d r for a form 7.
As an application, let S be a compact oriented submanifold of X ,of dimension p , without boundary, (or more generally a linear combination of pairs (Si,fi : Si -+ X ) as before). Then 6s is a current and d(6s) = 0. Let ws be a closed form on X which represents the Poincar6 dual cohomology class to the homology class of S (in singular homology of X).Considering the cohomology of currents on X , both 6s and w s represent the same cohomology class, and so their difference is exact:
+
where T is a current of degree n - p - 1 (or dimension p 1). However we will want to say more about being able to choose a T with desirable properties. Next we consider X as above together with a Riemannian metric. We then have operators d and d*, A = dd* d*d on currents as well as on forms (we define *, d*, in fact all operators on currents so as to agree with the definition of these operators on the subspace of differential forms. However currents are not given a pre-Hilbert space structure). The Hodge decomposition of A * ( X ) into the three subspaces Imd, I m C , 3c, which followed from the fact that if CY is orthogonal to X then there is a unique p with A@= a , in other words A * ( X ) = K @ AA*(X), A an isomorphism on XI,now says that if a current T annihilates 3c then it satisfies T = A S for a unique current S which annihilates X.This gives a decomposition of the space of currents into the same kinds of subspaces: Imd, Imd* , ker A. We now wee that a harmonic current has to be equal to a harmonic form considered as a current (since both map isomorphically t o cohomology). However a stronger result holds: The regularity theorem for the Laplacian A = dd*+d*d. To formulate this we have to define the notion of smoothness of a current T (on X ) on an open subset U of X.For non-compact manifolds such as U , one considers forms AE(U) with compact support in U as a
+
Iterated Integrals o n Compact Riemann Surfaces
71
subspace of A * ( X ) ,and defines currents on U in a similar way to current on X . Then currents on X restrict to (currents on) U . We say that a current T on X is smooth on U if there is a smooth form ru on U (not necessarily with compact support on U ) such that for all 4 E A: ( U ) ,T ( 4 )= 4 A rv (= 4 A T U ) . The regularity theorem now states that if S , T are currents on X and A S = T then for any open U , if T is smooth on U then S is also smooth on U . In particular if T is smooth on X then so is S , and if T = 0 on X then S is smooth and harmonic on X. Another consequence of regularity is the following: as before let 6s be the Dirac current of a cycle and w s a closed form representing the PoincarB dual cohomology class. Suppose T is a current such that
su
dT=6s-ws
and furthermore T is coexact (T = d*T'). Then: 1. T is uniquely determined by 6s and w s , and 2. T is smooth outside the support of S (i.e., outside the union of the images fi(Si) if S = C u i ( S i ,f i ) ) . To see this, note that if T = d*T' then we may assume T' E Imd, so T = d*dT1 and so dT
= dd*(dTl) =
(dd'
+ d*d)(dTl) = A(dT1).
Thus A ( d T l ) = 6s - w s is smooth outside the support S , so dT1 is smooth on this set and finally d * ( d T l ) = T is also smooth there. Finally, we may choose w s to be harmonic and then T , assumed coexact, is uniquely determined by S and by the metric on X and is coexact outside
S. If we do not fix a metric, then, given S , T exists but is not unique. The situation is better on a complex X with Kahler metric: we will use both d and d" there and the dd" lemma. The regularity theorem on a Kahler X is of course the same, but now the Laplacian A is A
= dd'
+ d*d = 16n2(d"d"* + dc*dc)
and the space of currents is the direct sum of 3c and 4 other subspaces, images of dd", d*d", d"*d, dc*d*. A regularity theorem for the operator dd" is : If a current T is dd" exact and T is smooth on U then T = ddCT1 for a unique current TI E Image of d*d"*, and TI is smooth on U . The proof is
72
Iterated Integrals and Cycles o n Algebraic Manifolds
the same as in the Riemannian case. To sum up, the regularity theorem on a Riemannian (compact) X implies all the other facts stated above. In the next chapter we will use currents T which are given by multiplying by an L' form T and integrating over the manifold X : that is T will be continuous on the complement of a submanifold S of lower dimension and, if T has degree p and a is any n - p form continuous on all of X then we require that a A T be an absolutely integrable n-form and T ( a )= (Y A T . We will say for short that T is given by an L1 form. We will also use Dirac 6 type currents 6s which are not L1. The main example of an L1 current will be the angular form around a submanifold S , gerenalizing d8/27r around the origin in the plane.
,s
Chapter 3
The Generalized Linking Pairing and the Heat Kernel
3.1
Introduction
In this chapter we will generalize some of the results of chapter 2 to higher dimensional Riemannian manifolds (always assumed compact oriented even dimensional); for compact complex Kahler manifolds our invariants will be independent of the choice of Kahler metric and depend only on the complex structure. We will begin with a Riemannian manifold X (as above) and a construction assigning to a pair of cycles A, B satisfying two further conditions, a real number ( A , B ) which is a generalization of the ordinary topological linking number (an integer). A, B are assumed to have dimensions p , q satisfying:
p
+ q = dimX - 1
(3.1)
Further, A and B must have disjoint supports, thus each of A, B is a formal linear combination (with integer coefficients) of differentiable maps of a standard simplex to X, and the support is the union of the images of these maps. The real number ( A ,B ) is obtained as follows: if A is the Laplacian on forms on X , then the heat operator exp(-LA) has a kernel K ( z , y , t ) ,an n-form on X x X for n = dimX (for each t > 0), and we construct a related
form F(z,y,t ) of degree n- 1and integrate I? over A x B , then take the limit as t 4 0. If A , B are both homologous to 0 in X then ( A ,B ) is an integer, the ordinary linking number (which is independent of the metric). If just A bounds, it follows that ( A ,B ) mod Z depends only on the homology class 73
74
Iterated Integrals and Cycles o n Algebraic Manifolds
of B and (A,B ) mod Z as function of the homology class of B , is just the Abel-Jacobi image of A in a torus. If A = dC and W B is the harmonic form Poincare dual to (the homology class of) B , then ( A ,B ) mod Z is given by - J, W B (mod Z). We will use the pairing ( A ,B ) also in situations where neither of A , B is homologous to 0. See [Harris, 19931. Let X have dimension n = 2m, and consider the heat operator exp(-tA) for all real t > 0, acting on forms (Y on X . Our main reference will be the book Heat Kernels and Dirac Operators by Berline, Getzler, and Vergne [BGV]. See also our paper [Harris, 19931. The operator exp(-tA) is given by a kernel K ( x , y , t ) which is a C” n-form on X x X x (R > 0):
in the notation of the appendix on orientations and fiber integration. As t + 0, this expression approaches a ( y ) .
K ( x ,y, t) is given by an infinite series if we choose an orthonormal basis of the forms A * ( X )which are eigenforms of the Laplacian A of X : denoting this basis a: with AaI = X(Y; and ordering the eigenvalues (for each degree p ) as 0 L XI L A2 5 . . . we get a series expansion:
where x, y E X , t > 0 , which is uniformly convergent, as are all x, y, t derivatives, because the eigenvalues tend sufficiently rapidly to m. Thus we can integrate term-by-term or similarly differentiate without worrying about convergence with respect to any of the variables x, y, t (for t > 0).
s’
By orthonormality of the (real) forms (Y: we mean (Y: A *a; = 0 if a: # a t and = 1 otherwise. The forms with X = 0 are a basis for the harmonic forms and the corresponding part of the series, denoted
(3.4) p
x=o
75
T h e Generalized Linking Pairing and the Heat Kernel
-
is the kernel for the projection onto harmonic forms. Also, H ( z ,y) = limit of K(x, y, t ) as t m.We will use d or d* to denote operators in the main variables ( 5 ,y). We can either use the series or appeal to the fact that exp(-tA) commutes with exterior differential d to see that K ( z ,y, t) is a closed form (for each t ) on X x X; also we can check that its DeRham cohomology class on X x X is Poincare dual to the diagonal. H is clearly a closed form as well, and Poincare dual to the diagonal. On X x X we use the product metric: then the Laplacian A on X x X is the sum A = A, +A, of the Laplacians on the individual factors. K ( s ,y, t ) satisfies a heat equation :
AK(z,y, t) = -2-
dK at
(3.5)
We can now give a formula for the coexact (on X x X ) form r(z,y, t) satisfying
Theorem 3.1 1 * r(x,Y,t) = 2 JI=t\d”K(? Y, 7Id7
(3.7)
where d* is the adjoint of d o n X x X . Proof: Write K ( z ,y, t ) - H ( z , y) = K ( z ,y, t ) - K ( z ,y, 00) = Jtrn(-%)d7=
;(AK)dr by the heat equation for K
But AK = dd‘K since dK
= 0,
so the last integral above is equal to
76
Iterated Integrals and Cycles on Algebraic Manifolds
Putting as the expression in square brackets, it is in the image of d* and CK= K - H , concluding the proof. 0 We will discuss the behavior of F(x, y, t ) as t decreases to 0 in detail below, and will just say here that r approaches pointwise a Cooform on the complement of the diagonal x = y. This limit form is singular on the diagonal but is L1 on X x X and I’approaches it while satisfying Lebesgue dominated convergence. On the complement of a neighborhood of the diagonal the approach of r to this limit is uniform. The limit form is just an “angular form” in the normal direction to the diagonal, that is, its integral over any small normal n - 1 sphere approaches 1 as the radius (distance to the diagonal) approaches 0. Once we have discussed the behavior of ( for t --+ 0) on the complement of a neighborhood of the diagonal, we can take the disjoint cycles A , B on X and form A x B on X x X so that the following expression makes sense.
Definition 3.1 r
The “angular form” lirnt+or then explains why ( A ,B ) is a generalization of the linking number . We will need some estimates on K ( x ,y, t ) and r ( x ,y, t): we may talk of these as functions rather than forms by using pointwise norms on forms given by the Riemannian metric. The main case to consider is when t is near 0 and x is close to y. We denote the square of the Riemannian distance by
dist(x,y)z = T ( Z , y)2 = T 2 .
(3.9)
The main fact is that for small t and r 2 , K ( z ,y, t ) is approximately the Euclidean heat kernel Q t ( x ,y) in R” x R”x R+, given by (for n = d i m X ) : Q t ( 2 , Y) =
(4rt)-n/z e -1z--yl2/4t ( d y l
-
&I).
. . (dy,
- dz,).
(3.10)
Appealing to [BGV] lemma 2.39 (pages 92,93) to bound the derivative d*X(x,y,t) we get:
The Generalized Linking Pairing and the Heat Kernel
77
For 0 < t 5 1, d*K(x,y , t ) is bounded uniformly on X x X by (const)t-(n+1)/2 (it suffices to see that if one differentiates the Euclidean Qt(x,y ) with respect to x or y , one calculates that dz(e-z2/t)= O(t-l/')). For t 2 1, any x or y derivative of K ( z ,y , t ) is bounded by a constant times e-"/' for X the smallest eigenvalue > 0 of A (BGV Proposition 2.36) and so the integral J,"O(d*K)dtis uniformly bounded on X x X . Putting these two bounds together we can write: roo
i
(3.11) Jt
2 1 and bounded by t(1-n)/2again uni-
is bounded uniformly in x , y for t formly in ( 2 ,y ) . For K ( x ,y , t ) we have bounds:
C2(47rt)-n/2e-r2/4tfor 0 < t 5 1 where r2 = dist(x,y)'. For r 2 E > 0, E fixed, K ( x ,y l t ) tends to 0 faster than any power of t , as t 3 0. Since these bounds are uniform on X x X I they persist after we integrate in y over a compact subset C of X : Lemma 3.1
Define (3.12)
(3.13)
Then I'c(x, t ) is O(t(1-n)/2)as t (distance x to C ) 2 E and t -+ 0 .
---f
0 , and K c ( x ,t ) is O(t-n/2e-E2/4t) for
Lemma 3.2 Let C l , . . . ,ck be chains in X (over which one can integrate forms) with k 2 3 and suppose C, . . . c k is empty (ie. the intersection of these k - 1 supports is empty). Then
n n
l?cl( 2 ,t ) A Kc2(X , t ) A . . . A KC, ( x ,t ) = 0
(3.14)
78
Iterated Integrals and Cycles on Algebraic Manzfolds
Proof: We can find E > 0 so that for each x E X,there is a Ci,2 5 i 5 Ic, such that dist(x,Ci) 2 6. Then Kci(x,t ) tends to 0 like e--e2/4t as t -+ 0, and the other factors are a t most tm. So the integrand is uniformly bounded on X and rapidly convergent to 0. 0 We need to extend this lemma to the case where X is replaced by a cycle C1 . . . Ci-1, assuming this intersection is defined and has the correct dimension - the main case being where C1,. . . , Ci-1 are submanifolds intersecting transversely.
n n
nCi-1 satisfying the condition above and C1n.. . nCi-1 nCi+l n. . . nCk being empty (i.e. the support is empty)
Lemma 3.3
With C1n.. .
we have:
Proof: Again there exists E > 0 such that each x E (support of) CI 0. . . Ci-lhas distance 2 E from a t least one of ci+l, .. . , c k l so the previous proof applies. 0
n
We need now information on the behavior of r ( x ,y, t ) as t 0 for (x,y) in a small neighborhood of the diagonal. Following [BGV], page 82 -+
and Theorem 2.30, write
r(x1d2 = 11t1I2
(3.16)
where E E T y ( X )and x = exp,(c) is the image of [ under the exponential map from Tvto a neighborhood of y. In Rn we could just write 5 = x - y. Also write N
k y ( x , y) = (47rt)-n/2 exp(-r2/4t)
tiGi(x,y )
(3.17)
i=O
for x near y and k r = 0 for ( x , y ) outside the neighborhood of the diagonal, (we are omitting factors ld~1'/~, Idyl1/' used in [BGV]), where @i(x, y) are C" sections (everywhere) of the bundle H o r n ( A * ( T i )A , *(Ti)) with G o ( z , x )= identity: we will write, using Iw" x Iw" notation,
The Generalized Linking Pairing and the Heat Kernel
79
Q0(z,y) = C(*dzr)A dyr modulo higher order terms in ~ ( zy), (where
I
denote multi-indices i l < . . . < ip). Theorem 2.30 gives the following pointwise, in (x,y), estimate: let N = (n/2) 1, then
+
d*K(z,y, t ) - d*k,N(s,y) = o p 2 ) uniformly for
( 2 ,y) E
(3.18)
X x XI as t --+ 0. Recall that:
r(z,Y1t ) =
r
- d * q x , y17)dT =
l+r
(3.19)
The second integral defines a C" function of (x,y) on X x X ([BGV] Proposition 2.37, page 93). We thus have to study s,' d * K ( z ,y, t ) , which estimate can be written as S,'[d*kT(x,y)]d7 0(1) by the above 0(t1l2) (as t -+ 0), (where we write kt for :k with N = (n/2) 1). Thus we have to calculate:
+
+
A. The Euclidean case, which is the "initial term" of d*kt, i.e.
(3.20) where Q(z,y, t ) is the Euclidean heat kernel Qt(z,y) =
Q(z,~ , t=) ( 4 ~ t ) - " / exp(-lz ~ - yI2/4t) x d(y1 - 21) A .. . A d ( ~ n ' -5").
(3.21)
B. The influence on A , from adding to Q extra terms Q.(xi -yi).'pi(z, y) or Q.tj.cpj(z,y) where j 2 1 and cp(z,y) are differentiable functions. We start with A and note that if we define
f
:IWn
XR"
f(z,y) = Y - z = E, then Q(z,y, t ) = f * I c ( [ , t ) where k([, t ) = ( 4 ~ t ) -e~p(-1[1~/4t)d[l ~ / ~ . . . den.
f * does not commute with d*
=-
* d*
because f * does not induce an isometry of T,'R with the normal to the diagonal at (x,5):
80
Iterated Integrals and Cycles o n Algebraic Manifolds
f * ( d [ i )= dyi - dxi has length 2, SO (3.22) Thus we have to calculate
To calculate d * k ( & , . . . ,tn,t ) we use the following notation: (3.24) where the ( n - 1)-form de denotes the unnormalized angular form around the origin of R" (rotation invariant form on any sphere S"-' with center 0) with
(I'(n/2) here denotes the gamma function, not the kernel). We let dB = r ( n / 2 ) ( 2 ~ " / ~ ) - ' & 9 be the normalized angular form on R" L' on R".
\ (0) (Jsn-,d0
= 1). This form is
We can now calculate (helped by [Hein]) d*lc(t,t ) = - *. d * [ ( 4 ~ t ) - ~ exp(/' C.$/4t)d
A
. . . A en]
* d [ ( 4 ~ t ) -exp(-r2/4t)] ~/~
= 2 ~ [ ( 4 ~ t ) - " / ~exp( - ' -r2 /4t)]rndB
Fix r2 # 0 and make the change of variable u = r 2 / 4 t , dulu = - d t / t , so
T h e Generalized Linking Pairing and the Heat Kernel
81
Note that the integrand is positive, and as t 4 0 with T~ fixed, the integral increases to e-'vn/2dv/v. As r -+ 0 this last integral increases to the (gamma function) r ( n / 2 ) . We conclude: if we pull back to R" x Rn by f*, $ d * Q ( s , y , r ) d r is bounded by the locally L1 form f * ( d e ) and as T 3 0 it approaches f ' d o (satisfying Lebesgue dominated convergence). Here f*dO is the angular form around the diagonal in Rn x R" and has integral 1 over every S"-l normal to, and centered on, the diagonal. de as n - 1 form on Rn is (up to a constant)
L1
[i/lcl
and since is bounded, its pointwise norm is l / ~ ~ - 'where T = I[\ , which is the norm needed so that its integral over any ( n - 1)-sphere is independent of the radius. If we were to multiply it by any positive power of T and by an everywhere continuous function, the resulting n - 1 form when integrated over an S"-l of radius T would approach 0 with T . On pulling back to Rn x R" by f * the same properties would hold in the normal direction to the diagonal. Such a multiplication of f*dO by r occurs in the asymptotic expansion k y ( x , y) in the initial term i = 0 when @o(z,y) is expanded in a Taylor series in powers of ( z j - yj) = <j (using coordinates in the tangent space Tyl i.e. geodesic coordinates centered at y), since as before (xj - yj)/r is bounded and x j - yj = r . (xj- yj)/r. The other terms, those involving t k @ k ( X , y) in the asymptotic expansion with k > 0, require a calculation of
which is almost the same as what was just done with Ic = 0 but with r - n / 2 replaced by T - ~ / ~ +The ~ .same change of variable v = r 2 / 4 r now gives the integral
82
Iterated Integrals and Cycles o n Algebraic Manafolds
so we can estimate such terms as bounded by the initial Euclidean integral times r 2 k . In conclusion, the asymptotic expansion of K ( z ,y, t ) with respect to powers o f t gives for r ( x , y , t ) a "leading term" which as t -+ 0 is bounded above by the Euclidean angular form f*dO (which is locally L1 and of order 1/rn-' in the normal direction to the diagonal, r being
the distance to the diagonal) and approaches this angular form as T -+ 0, with the rest of the terms after the leading one being of orders l / r n - k - l with k > 0. In particular, r ( z , y , t ) as t -+ 0 approaches an L1 form on X x X\ (diagonal) defining a current r(z,y) that satisfies
on X x X , A = diagonal, b~ = Dirac current, W A = harmonic Poincare dual form to the diagonal, r(z,y) = "angular current" . We want to see now that if the angular current I?($, y) for A c X x X is restricted to X x C for an oriented submanifold C of X and then integrated over y E C, one obtains an angular form for C c X I denoted rc. Using the asymptotic behavior argument, we are reduced to looking at X = R",n even, C = R"2 where n = nl n 2 and Wn = Rnl @ Rn2 is written as X = CL @ C (everything being in a neighborhood of the origin in Rn). C'- has coordinates (1c1 , . . . , zn, ,0 , . . . ,0) and normal bundle orientation given by WCI = dx,,+l A . . . A d x , (see appendix on orientations). C has the usual tangential orientation TC = w C l and normal orientation wc = ( - l ) n ' d x l A . . . A dz,, = ( - 1 ) " ' ~ c ~( S O TC A CJC = d ~ Al . . . A ds,). The angular form around C, d o c has, as current, differential d(d6'c) = b c = limit of Gaussian forms (Gaussian function of r1,t) x w c , where r: = + . . . x i , = square of distance to C. Thus d o c when restricted to CL = Rnl is (-1)"1d&, d& denoting the (normalized) angular form in Wnl around the origin.
+
XI
The Generalized Linking Pairing and the Heat Kernel
83
Consider now the commutative diagram
cl x c - x
ixl
x
c-x
f
(3.25)
where i = inclusion of C l to X, f(z,y) = y - z, p = orthogonal projection of to C'. Note that f o (i x I ) : Cl x C -+ X is an isomorphism but does not preserve orientation; however it does preserve the orientation in the second factor C alone, i.e. in the fibers of the vertical maps. Similarly, - p o i : C' -+ C' is multiplication by (-1) and so multiplies the orientation of C'- by (-1)"l. Now we recall that the angular form on X x X was f *dB and its restriction to X x C and image under prl, is p q * f * ( d B ) . Pulling back to C' by i* , we get i*(prl,f*dB) = prl,(i x I ) * f * d B (since the maps i x I, f map fibers C homeomorphically, preserving orientation of C), which is also = i*(-p)*p,(dB) = (image) p*(dB) on C' = Rnll under the map x -+ -x of R". It remains to show that p : R" -+ Rnl (orthogonal projection) sends d6' to p*(dB) = dB1 on CL = R"1. We can then conclude that i*(-p)*(dBl) = i*prl,(f*dO) = (-1)"ldBl on Cl,which is the restriction t o CL of the angular form d o c around C.
x
This last step for dB on R" = R"' @ R"* comes from factoring dB as follows, using T' = zf . . . xi = rf -I-r;:
+ +
s,'(l-
where B(nl/2,n2/2)= X)n1/2-1Xn2/2-1dX and dBi are the normalized angular forms of R"'. Then p*(dB) = integral of dB with respect to the variables xnl+lr...,xnl+nz = d B 1 J P J d & =do].
84
Iterated Integrals and Cycles o n Algebraic Manifolds
This concludes the discussion of I ' c ( X ) .
3.2 The main theorem We can now state and prove our main result of this chapter, expressing a generalized linking number by iterated integrals. This can be done on a (compact, even dimensional) Riemannian manifold but if the manifold is complex and Kahler then the real number obtained is independent of the choice of Kahler metric and depends only on the complex structure. Let Y be a smooth compact oriented Riemannian manifold of even dimension n, and let CI,. . . , ck,k 2 3, be smooth submanifolds of Y ,or more generally let each Ci be a cycle which is an integer linear combination of smooth submanifolds. Let Ci have codimension pi and assume Pi
+ . . . + P k = n + 1.
n
We will denote the support of Ci by Ci again and denote by the intersection of these supports. We make two assumptions: 1. The intersection of any Ic - 1 of the Ci is empty: c1 .rick = 0 for i = I,.. . ,IC. 2. The intersection of any set of Ci is transverse (that is, if Ci = C nijCij with Cij smooth manifold and if i l , . . . i, are distinct indices then Ciljl, . . . , CiTj,.intersect transversely). For simplicity we will write as if each Ci is an oriented manifold rather than a linear combination. Clearly 1. is the main assumption and we will see that 2. is not a real restriction. Let X = Y k= Y x . . . x Y with product Riemannian metric. i.e. { (y, . . . ,y)} be denoted Y , and let C1 x Let the diagonal in Y k , . . . x ck be the Cartesian product cycle in Yk.Then the generalized linking number
n...nci-,nci+,n..
(Y,c1 x ... x
ck)
(3.26)
is defined. It will be expressed using the following iterated integrals: let
wi be the harmonic (on Y)Poincare dual to the homology class of Ci. Then w1 A . . . A wi-1 is a closed form Poincare dual to the (homology class of) the intersection cycle of C1, . . . , Ci-1 , denoted C1 . . . 0 Ci-1. By the assumption l., this last cycle has disjoint support from the intersection cycle Ci+l . . . 0 ck.To include the case i = Ic in the notation, we define
T h e Generalized Linking Pairing and the Heat Kernel
85
= Y and write the two intersection cycles (with disjoint supports) as C 1 e . . .eCi-l and Ci+l . . . e C k + l for i = 1,. , , k (both being 0 for i = 1 ) . The closed form w 1 A . . A w i - 1 when restricted to the support of
Ck+l
.
.
ci+l
...
fying d q i - 1
ck+l
= w1 A
is exact: there exist forms q i - 1 on this support satis. . . A w i - 1 there. We then write:
n..
on this set Ci+l . n C k + l . Since w1 A . . . A w k = 0 we choose q k = 0 ( q k = 0 is all we need), Write q = Cr<s p T p , ( 1 5 r < s 5 k ) . We will need q only modulo 2: if j denotes the number of odd p i , then q f j(j - 1)/2 mod 2 ( j is odd since cpi = n 1 is odd). The result is then ([Harris, 20021):
sy
+
Theorem 3.2 With the above notation and for any choice of the forms q i - 1 (with q k = 0 ) , we have equality between the generalized linking number on X = Y k o n the left and iterated integrals on the right (recall k 2 3): k
(Y,ClX . . .x
c k )
=
(-1)q+'c/ i=2
.
[(WlA.. Wi-1,wi)-(w1A..
.AWi,
Ci+l*,.,*Ck+l
(3.27) (q = c r < s p T p s and the last term has ( w 1 A . . . W k , 1) = 0. This expression (either side of the above equality) is unchanged if any , Ci is replaced by a homologous cycle Ci satisfying the same intersection conditions, i.e either side of the equation is a function of the homology classes of the Ci; af any two Ci,say C,, C, for r # s are interchanged, the expression is multiplied by ( - 1 ) P r P s .
If Y is complex and the metric is Kahler then the expression is unchanged zf another Kahler metric on Y is used. Proof: We denote points in X x X as ( x , x ' ) . X = Y k and we write x = ( y 1 , . . . , yk), 2' = ( y i , . . . , y;). Since has product metric, its Laplacian A satisfies A = A 1 + . . . + A,, Ai = Laplacian on the ith factor Y , and the Ai commute as operators on X . Thus exp(-tA) is the product of the exp(-tAi) and the kernel K x for X is also a product.
x
K x ( 2 ,2',t>= P ~ K AY . . . A p r i K y = Ky(?4i, Y ; , t ) A . . . A X y ( Y k , Y k 1t )
I)]
86
Iterated Integrals and Cycles on Algebraic Manifolds
(we can also see this from the eigenform expansion). We abbreviate this as
K
= K 1 . . . Kk.
Similarly the harmonic part of K on X x X is H X = H = H1 . . . Hk. Recall that rx,t = r(z,x’,t ) satisfied two conditions: (a) d r x , t = K x - H x (b) rx,t is coexact (in image d*), If we change rx,t by adding an exact form
r>,t= rx,t + dE, then for cycles A, B on X with disjoint supports: SAxB ‘ i , t = JAxB
rx,t
-+
( A ,B , as
O’
Thus to define the linking number (A,B ) we can replace r by I” provided I” satisfies (a) and is orthogonal to harmonic forms on X x X . For X = Y k we can choose
.
.
= r1,tKz.. . Kk iHir2,tKs.. .Kk i. . iH I . . Hk-lrk,t
where ri,t= r ( y i , y:,t). The formula (a) for d r’ is easily checked since the Ki and Hi are even degree forms and d-closed. Orthogonality to harmonic forms on X x X is also obvious, since we can assume these to be products of harmonic forms on the factors Y x Y (corresponding to the eigenvalues 0 of A) and for each i the ith term of I?’ has a factor ri,t which is orthogonal to harmonic forms on the ith factor Y x Y . We now have to integrate F‘ = r ’ ( y 1 , . . . ,yn, y;, . . . ,y;, t ) over (yi, . . . , yk) E C1 x . . . Ck and ( y l , . . . ,yn) E diagonal of Y k . However the terms on the right hand side of the expression for I?’ list the y’s and y”s in the order y1, y;, .. . ,y n ry; and so we have to first move the yi to the right, leaving the y j on the left. In the j t h factor ( j = 1,.. . , k) of the i t h term we only need the differential forms of degree n - p j = dimension of Cj in the coordinates (yi, , . . . , yin) of yi , and so of degree p j in y j coordinates (for the factor Hj or K j ) or p j - 1 in y j (for the factor rj).Thus in the ith term moving all dy‘ to the right introduces the sign (-1)Q+ni
where q = C r < s p r p s and IT^ = El-’ p j (since n - p j = p j mod 2).
The Generalized Linking Pairing and the Heat Kernel
87
We can now integrate the ith term over CI x . . . x c k . According to the Appendix on Orientations] this means that we integrate the j t h factor over yi. E Cjand multiply the resulting forms in the variables y j . By the lemma in this appendix] this integration] denoted p r l , I gives prl,(Hj) = wc, = w j (harmonic Poincare dual form to Cj in Y) prl*Kj = KC,
where drc, = KC, - WC, on Y (rc,and KC, also depend on t > 0). Thus denoting prl : X x X + X the first projection, we have: k
c ( - l ) q + T ' u w l ( y i ) A . .. ~ W ~ - i ( ~ Z - i ) A \ r c , , t A K c , + l , , A.AKcL,t. ..
pTi*r\,t
a=1
(3.28) Next this has to be restricted to the diagonal Y : y l = . . . = Yk = y of Y k l giving the wedge product on Y, and finally it has to be integrated over y E Y, giving (Yl c 1 x . . . c k ) when the limit t + 0 is taken. This last integral is then k
sy n
where L J ~= W C , . In this sum the first term is (-1)q J7Cl,tKCz,t... KCk,t which by lemma 3.2 approaches 0 as t 4 0 (since C2 . . . c k is empty). For i 2 2 we write C1 . Ci-1 = Cl..,i-l and wl...i-l for the Poincare dual harmonic form. Thus
n.. n
wl..,i-l
= w1 A
. . . A wi-1
+ dai-1
(ai-1 =
smooth form) .
Also we have, as currents, W l ...2 - 1
= &!I ...2-1 - Dl ...2-1
n n
where I'l. ..i-1 is smooth on the complement of C1 . . . Ci-1 and in particular is smooth on Ci+l . . . c k c k + l which we define as X for i = Ic (we take c k + l = X and for i = k, r l . . . k -1 is smooth on X ) . Thus
n n n
Iterated Integrals and Cycles o n Algebraic Manajolds
88
and the ith term in the integral is
(-
l)P+Ti
s y bC1 ...i-1
(-1)QfTi
Jy
A
rci,t KC,+lJ l l
d ( r 1...i - 1 +
*
a
A
*
KCk,t
-
a i - i ) r c i , t K c i + l , t ..Kck,t. .
The first of these two terms is, by definition of the Dirac current b,
* ,-,...n ci-lrci,tKci+l,t. . .KCkrt SC,
n n
n
n n
and since C1 . . . Ci-l Ci+l . . . c k is empty this term is 0 for i = k and for 1 < i < k it is exactly the same as for i = 1 but with X replaced by Cl 0,.. nCiP1;thus as t -+ 0 this term approaches 0. Thus we are ai-1) is a current left with the second of the above terms: here d ( r i...i-1 and the meaning of the integral (including the sign in front), since degree of l?l...i-l + ai-l is 7ri - 1, is:
+
(-l)qslJ y ( r 1.,.i - 1
Write
rl,,,i-1 -tai-1
(-1)'Jy
Vi-lKci,t
A
+ ai-l)(drc,,t)
Kc,+l,t A KCk,t.
= -qi-l, then this integral is a sum of two terms:
. . . A Kck,t - (-1)'
Jy V i - l W i A
Kci+l,t A . . . A KCL,t.
n n
In the first integral, vi-l is an L1 form, the angular form for C1 . . . Ci-1, and is smooth outside a neighborhood of this set (which does not meet Ci+l . . . Ck). The integral over this neighborhood approaches 0 as t --+ 0 since Kci,tA KC,,^,^ A . . . A KCk,tapproaches 0 rapidly here. ~ 1 . . . i - 1 being smooth over the complement of this neighborhood, which includes Ci . . . C k on which Kc,,t . . . KC,,^ approaches ~c,...c,,this integral vi--1. then approaches ( - l ) q In exactly the same way, but using disjointness of C1 . . . Ci-1 and Ci+l . . . c k , the second term approaches
n n
n n
sci,,,ck
n n
n n
-(-I)'
Jci+l,,,ck Vi-lwi.
In each of these integrals, qi-1 is a smooth form on the domain of integration which is a manifold (or linear combination of manifolds) and wi, Poincare dual to in is also Poincare dual to . . . c k in ci+l . . . c k (by our transversality hypothesis). By Poincare duality in Ci+l 0 . . . 0 ck,if cp is a closed form on this manifold (or linear combination) then
ci x,
ci
sci...ck 'p - sci+l...ck cp A W i
= O.
is not a closed form: instead dqi-1 = w1 A . . . A wi-1 on Ci+l . . . c and the above argument with closed forms cp says that
qi-l
k ,
The Generalized Linking Pairing and the Heat Kernel
sci . . . C k% - I
89
- sCi+l.,.Ck Vi-1 A W i
depends only on d v i - 1 , i.e. only on wl,. . .LJi-1 and ci+l,. . . , c k . We can then write the above difference of two integrals as
Jc.*... C,(WI A . . .
AWi-lrl)-sci+l,,,Clr(W1
A...Awi-l,Wi).
We recall that w 1 A . . . A wk = 0 (its degree is higher than n = dimY) and we choose q l , , , k = 0. Now adding up the k. integrals, the first being 0, we get the expression in the theorem (the last term in the sum being (-1)"'
S,(Ul
A . . .A Wk-l,Wk))
.
If k = 3 we find: for C1, CZ, C3 mutually disjoint, (YlCI x c z XC3)=
(-l)"f"~c,(W1,W2)-(W1AW2,1)+sy(~1AWz,W3)l.
We note now that the iterated integral expression does not involve the cycle C 1 , but only involves its harmonic Poincare dual w 1 . The iterated integrals are over Ci+l . . . ck+l for i 1 2. We conclude that the linking number as well depends only on the homology class of C1 (and on the metric). Next we will prove a (super) symmetry of the linking number under interchanges of the Ci, which will also prove its dependence only on the homology classes of all the Ci. We will only give details for interchange of C 1 , CZ,that is,
(Y,c2 x
c1 x c3 x . . . x c k ) = ( - l ) p * p(Y, zc 1 x cz x . . . x ck)
(3.30)
(recall pi = codim(Ci)). To do this we just note that instead of carrying out the integration with the form r>.tabove, we can use
Then
and the same calculation as before gives this as
Finally we assume X is complex with Kahler metric and examine the effect of changing to another Kahler metric with the same complex structure.
90
Iterated Integrals and Cycles o n Algebraic Manifolds
Denote with ' quantities depending on the second metric. Then wi is harmonic in the original metric and w: is the corresponding harmonic form in the second metric. Then dwi = 0 = dCwi and dw: = 0 = d"w: (d,d" depend only on the complex structure) and wl - wi is d-exact. Since wi - wi is also d" closed, the dd" lemma says that there is a form X i on Y satisfying wi - wz = dd"Xi with i = 1,.. . , Ic (recall dd" = (i/27r)L@). Thus W;
A
. . .W :
= W1 A..
. A W i + dd" Ci=1~1 . . .wj-1Xjw;+i . . .W : .
n n
Having chosen any qi on Ci+2 . . . Ck+l satisfying q k = 0 or at least J y q k = 0) we can choose: 7; = qi
. On c k + 1
dqi
...
+ d " C j = , w l . . . ~ j _ 1 X j ~ ~ + l w:.
= X we get from q k = 0 that
s,
and so q; = 0 since Y is a complex manifold. For i 5 k, start with
and multiply by w: = wi
+
dd"Xi,
getting
Subtracting this from the equation for q:, we have
Since dcdqi-1
= d"w1
. . . wi-1
= 0, this is
and so the integral over a cycle, e.g.
Ci+l
. . . c k + 1 is 0:
= w1
. . .w, (and
The Generalized Linking Pairing and the Heat Kernel
91
This concludes the proof of the theorem. 0 The following question now arises: is it possible to replace the disjointness hypothesis by a hypothesis on products of cohomology classes? Even the answer to the following basic question seems unknown: if X is a compact oriented manifold, simply connected, and Kahler, and a , p are cohomology classes with real coefficients whose cup product is 0, can their Poincare dual homology classes be represented by cycles with disjoint supports? We will now state a “complex version” of this last theorem. X is again a compact complex Kahler manifold and K ( z ,y,t), H ( z ,y) are the heat kernel and the kernel for orthogonal projection to harmonic forms. y(z, y, t ) is the coexact form on X x X (for each t > 0) satisfying:
y is in image of (dd“)”, and
(thus if X has complex dimension 1 then y is of type (1 - 1 , l - 1)). For some formulas and details on y,see [Harris, 19931 and [Harris, 20021 . Let (A,B ) be complex-analytic cycles on X which are disjoint and whose complex dimensions satisfy
dimCA + dimCB = dim@X- 1 . We define the Archimedean Height Pairing ( A ,B ) as
(3.31) If we integrate y over A first we obtain a form y ~ ( yt,) and as t -+ 0 one can show that y ~ ( yt ,) approaches a current YA which is a smooth form outside of A and so can be integrated over B. Such currents are sometimes called “Green’s forms” for A . However we need not elaborate on this and just use
r(x,Y,t ) . Suppose now X = Y k = Y x . . . x Y where Y is complex Kahler of complex dimension m and c1,.. . , ck (k 2 3) are compact complex submanifolds of Y (or cycles which are linear combinations of submanifolds) and satisfy the following conditions as cycles in Y :
Iterated Integrals and Cycles o n Algebraic Manifolds
92
+
k
codim(Ci) = ( d i m Y ) 1 (all dimensions and codimensions are (a) complex). (b) Any k - 1 of the Ci have empty intersection. (c) All intersections are in “general position”: if each Ci as cycle is Ci = C a i j C i j (Cij being submanifolds of Y ) , then for any distinct i l , . . . , i,, ciljl, . . . ,Cirjr intersect in general position (and so the intersections are submanifolds). Let a l l . .. , a k be the harmonic Poincare dual forms to the homology classes of the Ci. Condition (b) implies that for i = 2 , . . . ,k there exist forms pi-1 on Ci+l 0.. . c k such that a1 A . . . A = ddCpi-l on this intersection. We set p k = 0. (For instance, we could use the dd“ lemma, or else take pi-1 = -rz,where 2 = C1 0... 0 Ci-1). For any choice of such pi-1, we write
n
We then have
Theorem 3.3 Let X = Y k with product metric ( X , Y are compact Kahler). Let c1,. . . ,c k be complex cycles on Y satisfying conditions a , b , c above, and let ( A , B ) denote the Archimedean pairing on X . Let C1 x . , . x c k be the product cycle on Y k and Y be the diagonal cycle. Then k
(Y,c1x . . . X C k ) =
.
(a1A . . A a i - l
i=2
c,+1 . . ..Ck+
, ai)- (a1A. . .A a i , 1)
1
(3.31) (here c k + 1 = Y ) . This real number (either side of the above equation) depends only on the homology classes of the Ci, and is unchanged under permutation of C1, . . . , c k .
Proof: The proof is essentially the same as the proof of the previous theorem involving real cycles (but with fewer or - signs as all dimensions are even). However the last statement of the previous theorem (independence of the metric) is not known to us to be true here since the proof does not carry over. We remark that in the case where all the Ci are divisors we can choose as Green’s currents yci = log (Ioill’,ui being meromorphic sections of
+
The Generalized Linking Pairing and the Heat Kernel
93
metrized analytic line bundles on Y (with the metric 11 11 normalized so that dd'yc, = 6ci - aci, aci = ai harmonic as before). Comparing the calculation of the height pairing above with [Deligne 1987, Sec. 81 we see that he is studying the same height pairing for divisors Ci. Deligne does not assume the intersection condition (b) above; consequently our statement on homology classes does not hold in his situation.
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Appendix: Orientations, Fiber Integration
It is necessary to orient a compact n-manifold in order to integrate an n-form over it. We make explicit here how we make various choices of orientations . All manifolds will be C”. Let X be a compact oriented n-manifold and S a compact k-dimensional submanifold. An orientation of S is equivalent to an orientation of its normal bundle N = Ns by the following convention: a t x E S we choose an ordered basis of the tangent space Tx(S) in the orientation class of S and follow it by an ordered basis of N,. The resulting ordered basis of Tx(X is required to be in the orientation class of Tx(X). Our Poincare duality convention is then as follows: let ws be a closed form on X such that for every closed form LY on X we have Js~ = Jxa !A ~ s.
Such ws exist and will be called “Poincare dual forms to 5’”. For instance, w could be a Thom form, Gaussian shaped in the normal direction to
S
and defining the normal orientation to S. Further, w s could depend on a real parameter t > 0 and as t -+ 0, w s could approach the Dirac current 6s of integration over 5’: this means that for all forms (Y (not necessarily closed) on X we have:
Jx
a!
A ws
4
Js a! (written also as Jx a! A 6s)
as t 4 0. Next we consider two oriented submanifolds S1,S2 of X which intersect transversally so that their intersection T is a submanifold of dimension dimT = dim& dim& - n. We orient T so that an oriented ordered basis for the normal bundle to T in X consists of an oriented ordered basis for the normal bundle to T in Sz (which then also orients the normal bundle
+
95
Iterated Integrals and Cycles on Algebraic Manifolds
96
to S1 in X ) followed by a basis orienting the normal bundle to S2 in X . Poincare dual forms w s l , ws2 t o S1,Sz in X then determine a Poincare dual form w T : WT
= US1 A WS2 .
Also, the restriction of wsl to S2 is a Poincare dual form to T in S2. We also have to orient Cartesian products of manifolds and of submanifolds. If X 1 , X2 are oriented manifolds of dimensions n 1 , n ~we write p r i : X 1 x X 2 -+ X i for the projection, i = 1 , 2 , and for a1,a2 forms on X I ,X z , we write pr;crl or a(x1) and prZa2 or C Y ~ ( X Z for ) the corresponding forms on X1 x X2. If X I ,X2 are oriented tangentially by top degree forms w1, w2, we orient X 1 x X2 by p r i w l Apr,*wz. So
sxlxx2
(Pr;771 A P a r l a ) = sx,771
sx,
772
.
Let A l , A2 be oriented submanifolds of X I ,X2 of dimensions a l , a2 and with Poincare dual closed forms W A ~W, A ~ . We would like to define the orientation and Poincare dual form W A x ~ ~ of A1 x A2 so that for any closed forms ai on X i , d e g a i = ai, SA1 x Az prrcrl A
prlaz =
sA1 sA2 01
a2
=
sxl
xx2
p r ; a A pr,*az A W
A X~A ? ,
For this we have to take WA1x A 2 = (-1)(n1-a1)a2pr;WA1 A pr,*wA2
.
In particular, for X 1 = A1 we get w X 1 x A z = pr$wA2
For X Z = A2 we get: W A l x X z = (-1)(n1-a1)n2pr;wAl
.
Multiplying, W X i X A2 WAl X X Z = WA1 x A ~ .
For n even W A x~ ~ =z ( - l ) a 1 a 2 p r ; W A 1 A praWA2. As a n example we consider in X x X the diagonal A (A should not be confused here with the Laplacian). Using local coordinates 51,. . . , z, in X , X is oriented by the n-form dxl A . . . A dxn. We take a copy Y of X with coordinates y 1 , . . . , yn (p*yi = xi if p is the identification map p : X + Y ) and write the coordinates in X x X as ( X I , . . . , x,, y1,. . . ,y,). We use the
2
Orientations, Fiber Integration
97
diagonal map i : X -+ X x X , i(z1,.. . ,z,) = ( 5 1 , . . . , x n , x l , .. . ,x,) and orient X and A so that i is orientation preserving as map from X to A. We orient X as above and orient the tangent bundle of X by the ordered basis a , the tangent bundle of A by the basis . . . , aaz, da~ , a . . .,K and the normal bundle of A in X x X by . . ,- K a Then a a a X x X is oriented by . . , =, . . . , The projections p r j , = 1 , 2 , of X x X to X then induce orientation preserving diffeomorphisms of A with X . Let now A l , A2 be oriented manifolds of dimensions kl , k2 and let f i : Ai 4 X be differentiable maps. We assume that f l x f 2 : A1 x A2 + X x X is transverse to A c X x X . For example, this transversality holds if A1 = X , f l = Identity. Then the fiber product of fl x f 2 with the inclusion of A, in other words the inverse image of A in A1 x A2, is a submanifold A1 x x A2 of A1 x A2 with normal bundle in A1 x A2 oriented by ( f l x fi)*(wa)where WA orients the normal bundle of A in X x X . Assume now A1 = X , f 1 = Identity map I , write F = I x f : X x A2 + X x X . Then F-l(A) = graph o f f = ( ( f ( a z ) , a z ): a2 E Az}, denote this by g r ( f ) . We claim: A2 + gr(f), induced by F , is an orientation preserving diffeomorphism (taking a to ( f ( a ) , a ) . To see this, we take any top degree form p on A2 that orients A2 and show first that under pr2 : gr(f) -+ A2, we have that pr;p is a top degree form on gr(f) that gives an orientation of g r ( f ) . Namely, if the tangent space to A2 at a point a2 has oriented basis , . . .&, k = dirn(A2),then a t ( f ( a z ) , a 2 ) ,g r ( f ) has tangent space basis ( f * ( € ~ @ ) & , . . . , f*(&) @ &). Assuming p(C1 A . . . A &) = 1, we find that
&& + &-, + -&+&. +&. =, x.
=,
&:.
Pr,'(P)[f*(
Next we recall that
WA
1.
@
on X x X is locally
(-dxl
+ dyl) A .. . A (-dx, + dv,)
We can then calculate that on
.
x x A2, if p = 01 A . . . A p k ,
Now assume n is even, obtaining
dzl A
. . . A dx,
A
f*@)
98
Iterated Integrals and Cycles on Algebraic Manifolds
which is the orientation of X x A2. In conclusion, the orientation of gr( f) by ( I d x f ) * w a agrees, under the map A2 + gr( f), with the orientation of A2. As our main calculation we will state the following lemma in which we use integration over the fiber prl*: this integration will be reviewed after the statement of the lemma. We denote by f * d A 2 the current on X image of d ~ on , A2.
Lemma. As before denote by F the map I d x f : X x A2 + X x X and by g r ( f ) the inverse image F - ' ( A ) C X x A2. Let prl : X x A2 4 X be the projection and prl* : AP(X x A2) + Ap-'(X) (Ic = dim(A2)) (degree p currents on X x A2) --f (degree p - k currents on X ) the integration over the fiber map. If W A is any closed form on X x X , Poincare dual to A, then prl,(F*wA) is a closed form W A ~o n x , Poincare dual to the cycle f*(A2) an X . If kt,A for each t > 0 is a Gaussian-shaped Thom f o r m on X x X which as t -+ 0 approcahes the current 6~ then F * ( k t , a ) is a similarly shaped form on X x A2 approaching the current dgr( f) as t 4 0 , and prl.F*(kt,A) = k t , ~ approaches , f * ( d A , ) on x as t + 0. Finally, if r ( x ,y, t ) as f o r t > 0 a smooth f o r m on X x X satisfying d r = kt,A - W A then rt,AZ= p r l , F * ( r ) is a C" f o r m on X satisfying n t , A Z = kt,A2 -wA,. Proof: 1.To prove the statement about W A , let a be any closed form on X , then Jx a A p r l * F * ( u A )= & x ~ z P r ; aA F * ( u A ) = . f g r ( f ) p 6 a (since F * ( ~ Ais) a Poincare dual form to gr( f)). Now using the orientation preserving diffeomorphism f x I d : A2 + gr( f) and the map pr1 o ( f x I d ) = f : A2 3 X we get: S g r ( f ) p r T a= s A , ( f Id)*pr;a = J A , f * a = f * ( d A z ) ( a ) .
2. For any form a on
X,
Jx a A P r b ( F * k t , A )= J X x A z Pr;a A (F*kt,A). Write F * ( k t , A ) = I c t , g r ( f ) : since F is transverse to A, kt,gr(fl is a Gaussian shaped form on X x A2 peaking on g r ( f ) . Thus the last integral above approaches J g r ( f ) p r ; a as t -+ 0, and as in l., this equals J A f*(a) ~ = f*(dAz)Q*
3. Starting with dI'(x, y, t ) = kt,A - W A on X x X we apply F * and then prl*, both of which commute with d, obtaining the statement for rt, kt,Az, wA2 on x . 0
Orientations, Fiber Integration
99
Integration over the fiber Denote by X I , X z compact oriented manifolds of dimensions n1,nz and by p r i : X1 x X 2 -+ X i the projection. Denote by A* the differential forms on these manifolds. We define: :A*(X1 x
X z ) -+ A * ( X 1 )
in such a way that for ai E A * ( X i ) ,
pri* (pr;ai A P r l 0 2 ) = a1
(sx,
az)
[similarlyprz+(pr;al A prlaz) = (J,, a1)az]. Then for cp E A * ( X 1 ) ,$ E A*(X1 x X Z )
Prl*[(Pr;cp)A $1 = cp A P T l * ( $ )
J,,
x x z[(Pr;cp)A $1
E
A*(Xl)
= sx,['p A P T l * ( $ ) I .
If X I ,X2 have no boundary then d o prl, = prl* o d
d oprz* = (-l)n1pr2* o d
(nl
= dimX1).
We extend prt to a map from currents Ti on X i to currents p r f ( T i ) on X1 x X2 in such a way that if Ti is given by a form ~i on X i (thus Ti(ai)= oiA T ~ then ) prf is given by the form p r t ( q ) . Thus we define, for ai E A * ( X i ) ,
I,,
Pr;(Tz)(Pr;W
APT;~Z= ) (JX,
[in particular, p r l ( 6 ~=~ S)
~I)Tz(~z) = Tz(P~z*(W;WP G W ) )
X ~ ~ and A ~ ]
pr;(Tl)[pr;al A p r , * a ~=] ( - l ) ( d e g T l ) ( d e g a 2 ) T ~ ( a l )
(assuming deg
[In particular pr;6A1 = xxz .I We thus have pr;(Ti) = Tio p r i , : A*(X1 x X z ) -+ Iw for all n1 and even nz. For a general map f : X -+ Y we define, for a current T on X , form a on
y,
f*(T)(a) = T(f*a).
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Bibliography
NOTES We would like to point out in particular the following references: 1. K.T. Chen’s “Collected Papers” and the summary in it of his life and work by R. Hain and P. Tondeur. 2. The survey article by R. Hain, “Iterated Integrals and Algebraic Cyles: Examples and Prospects” in the volume ‘‘Contemporary Trends in Algebraic Geometry and Algebraic Topology” edited by S.S. Chern, L. Fu and R. Hain (Nankai Tracts in Mathematics voi. 5, World Scientific, 2002), and many other valuable papers by R. Hain. 3. The book “Heat Kernels and Dirac Operators” by N.Berline, E.Getzler, and M. Vergne. 4.The article “Groupes Fondamentaux Motiviques de Tate Mixte” by P. Deligne and A.B. Goncharov, giving some different viewpoints and directions from those in this book. 5. The survey article “The Ubiquitous Heat Kernel” by Jorgenson and Lang. 6. Many articles by R. Harvey and B. Lawson on currents, differential characters and singularities.
THE REFERENCES Berline, N., Getzler, E., Vergne, M. (1992). Heat Kernels and Dirac Operators. Springer, Berlin. Bloch, S. (1984). Algebraic Cycles and Values of L-functions. J.Reine.Angew.Math. 350, pp. 899-912. Chen, K.T. (2000). Collected Papers of K.T.Chen. P.Tondeur, Editor. Birkhauser, Boston. Deligne, P. (1987). Le Determinant de la Cohomologie. Contemporary Mathematics 67, pp. 93-177. Deligne, P. et Goncharov, A.B. (2003). Groupes Fondamentaux Motiviques de Tate Mixte. ArXiv:Math,NT/0302267. Griffiths, P. and Harris, J. (1978). Principles of Algebraic Geometry. Wiley. Hain, R. (2002). Iterated Integrals and Algebraic Cycles: Examples and Prospects. In: Contemporary Trends in Algebraic Geometry and Algebraic Topology, Editors S.S. Chern, L. Fu,R. Hain. World Scientific. 101
102
Iterated Integrals and Cycles o n Algebraic Manifolds
Harris, B. (1983). Harmonic Volumes. Acta Mathematica 150, pp. 91-123. Harris, B. (1983). Homological vs Algebraic Equivalence in a Jacobian. Proc.Nat.Acad. of Sciences USA 80, pp. 1157-1158. Harris, B. (1989). Differential Characters and the Abel-Jacobi map, pp. 69-86. Algebraic K-Theory: Connections with Geometry and Topology, Ed. by J.Jardine and V.Snaith. Kluwer. Harris, B. (1990). Iterated Integrals and Epstein Zeta Function with Harmonic Rational Fucntion Coefficients. Illinois J.Math 34. pp. 325-336. Harris, B. (1993). Cycle Pairings and the Heat Equation. Topology 32. pp. 225238. Harris, B. (2002). Chen’s Iterated Integrals and Algebraic Cycles. pp. 119-134. In: Contemporary Trends in Algebraic Geometry and Algebraic Topology, Ed. by S.S. Chern, L. Fu, and R.Hain. World Scientific. Hein, G. (2001). Computing Green Currents via the Heat Kernel. J.Reine und Angew.Math. 540, pp. 87-104. Jorgenson, J. and Lang, S. (2001). The Ubiquitous Heat Kernel, pp. 655-682. In: Mathematics Unlimited: 2001 and Beyond, Ed. by B. Engquist, W. Schmid. Springer. Kodaira, K. and Morrow, J. (1971). Complex Manifolds. Holt, Rinehart and Winston. Lazard, M. (1954). Sur les Groupes Nilpotents et les Anneaux de Lie. Ann.Ecole Norm. Super. 71, pp. 101-190. Quillen, D.G. (1968). On the Associated Graded Ring of a Group Ring. Journal of Algebra 10, pp. 411-418. Schiffer, M. and Spencer, D.C. (1954). Functionals of Finite Riemann Surfaces. Princeton University Press. Weil, A. (1962). Foundations of Algebraic Geometry (2nd edition), pp. 331. American Mathematical Society. Weil, A. (1979). Collected Papers. Paper 1952e and Comments. Springer.
List of Notations
I
integral, iterated integral 1 fundamental group 1 commutator subgroup terms of lower central series 1 ((n, n),. . . ), n) A Laplace operator on forms 2 ,-tA Heat operator 2 G Lie group 2 0 Lie algebra (of G) 2 TdG) tangent space at e 2 Ai(X) differential i-forms on X 3 CY 1-form, with values in g 3 1% PI bracket of Lie algebra valued 1-forms 3 U Maurer-Cartan g-valued 1-form on G 3 7 Chen's connection: a g-valued 1-froms on a manifold X e A, reduced diagonal map: H 2 ( X ) -+ H l ( X ) @ H 1 ( X ) 7 Rn1 group algebra of n1 7 7 I,E Augmentation ideal in Rn (kernel of Rn -3 R) Gr(Rni) associated graded algebra: @F'-o((IW.rr)n/(G)n+l 7 7 n1 3 (7r1,Tl) 3 . * . lower central series of a group n1 U(e) universal enveloping (associative) algebra of a Lie algebra 0 8,9 iterated integral homomorphisms given by 11 Chen's connection 8 8, 15, 19, 21 Q Quillen homomorphism U(Gr7r18 R) -+Gr(Rn) 8 , 20 function n -+ E, q ( y ) = y 1 20 4 tensor algebra T ( V ) R1@ V @( V @V )@ . . . (free associative algebra) generated by a vector space V 9 A diagonal homomorphism H -+ H @ H of a Hopf algebra H 9,lO n1
1
103
104
Iterated Integrals and Cycles on Algebraic Manifolds
L free Lie algebra 10 H Hopf algebra 10, 11 P(H) primitive elements of a Hopf algebra H 12 completions of the graded Lie algebras L , g L"10" (formal infinite series) 12 all elements of degree L n, g" = l@(g/g,n) L>n10>n 12, 13 A extension of A : U ( g ) 4 U ( g ) I8 U ( g ) to a homomorphism U(dA ( U ( 8 ) U(€l))A 13 B(n) B / B > n (Lie algebra) 15 % Lie group of ( g / g > n ) 15 h(n) group homomorphism T I -+ G(n) 15 IT same as i E 16 Xi, C 1 ,E2 special subspaces of the differential forms A i ( X ) 16, 34 XP harmonic pforms 17 Q Lie algebra homomorphism induced by q 20 ,(a, ,(n) ( T , T ), (dn-l) , T ) terms of lower central series 20 B ( T ) ,K(T11) classifying space of a discrete group T 22 Ti i-th homotopy group 22 image of 7r2 ( X ) -+ Hz ( X ) ,spherical homology classes 23 HiPh( X )
"
+
Tp
Tl/(Tl,
m)
Ai
i-th exterior power 25 41 11, Lie algebra homomorphisms 25, 26 g[nl " e 1 , L l 1 I 1 . . . ,s11(nfactors) 27 ($4 exp g In] 9 Riemannian metric 29 d vol volume element 29 * Hodge star operator A P ( X ) -+ An-p (4 29 d* adjoint of d 30 A Laplace operator:dd* d*d (sometimes denotes diagonal). P(Y) orthogonal projection of y in A p ( X ) to XP 30 almost complex structure operator, T , ( X ) ---f T , ( X ) J 4% w) Kahler 2-form 32 dC operator &(J-' o d o J ) on forms, also = i(8 - a)/47r (dC)* adjoint of d", also = J-'d* J/47r 33
+
e
simplified version of 0
35, 36
Xi harmonic 1-forms with periods in Z 36 (Xic3 Xi)' kernel of intersection number pairing
37
30 31 33
List of Notations
105
, az;7)
iterated integral over y 37 homomorphism defined by I(a1, 0 2 ; y) 37 (Xi8 Xi 8 Xi)' a subgroup of (Xi)@' 37 (A3Xi)' a subgroup of A3Xi 38 Ail Bi, C i simple closed curves 38, 40 P same as (A3%;)' (later) Lefschetz-primitive cohomology classes 40, 46 v Y = 2 1 : P --f R/Z 40 A Alel- Jacobi map 42 D a 3-chain in R3/Z3 42 Jacobian of a Riemannn surface X 45 J a c ( X ) ,4x1 Ximage of A ( X ) ( i e . , X C J a c ( X ) ) under map g + g-' of J a c ( X ) 45 W Kahler 2-form on J a c ( X ) 45 S Schiffer-Spencer coordinates (s1, . , ~ 3 ~ 4 ) 48 Q map P @z JR + H o ( X ;K 2 ) 49, 50 j3 complex structure 41 T current 67 6, Dirac current 67
I
(4B )
Real valued linking number 73, 76 exp( -tA) Heat operator 73 K ( z ,Y,t ) l K kernel of the Heat operator 74 75 form satisfying dI'= K - H r(z1Y,t ) ,r eigenform of A 74 76 square of Riemann distance dist(z,Y)2, d z , Qt (5, Y) Q(z,Y,t ) Euclidean heat kernel 76 ci chains, usually cycles 77, 78, 84 K c , rc forms attached to C 77 asymptotic expansion of K ( z ,y, t ) to order N k,"(XI Y) dB (unnormalized) angular form around origin in R" dl3 normalized angular form 80 Pr* integration over the fiber of pr 87, 98 c 1u . . . u c i intersection of supports (sets) 84 c1. ... . c a intersection cycle 84 (wlA*..AWi_l,~i) vi-1 Awi where dqi-1 = w1 A ' . . A w i - l (w1 A . * .A ~ i - 1 , l ) vi-1 85 86 K11..' 1Kk K i = p r f K ( z 12/, t ) = K ( z i ,y i , t )
4
78 80
85
106
Iterated Integrals and Cycles o n Algebraic Manijololds
H i 1. . . Hk Hi = p r f H ( x ,9) = H ( x ~yi) , Y(X1Y't)' Y dd"7 = K - H 91 ?A Green's form for complex cycle A
86 91
Index
L’ current (and L1 form), 72 d,d‘ lemma, 34
harmonic forms, 30 harmonic volume, 42 heat equation, 75 heat kernel, 74 heat operator, 73, 74 Hodge *-operator, 29 Hodge theory, 28 Hopf algebra, 10, 11 hyperelliptic Riemann surfaces, 50
Abel-Jacobi map, 42 algebraic equivalence of cycles, 52 angular current, 82 angular form, 76, 80-83, 88 Archimedean height pairing, 91 asymptotic expansion, 78, 82
integration over the fiber, 99 intermediate Jacobian, 46 iterated integral, 1
Chen’s connection, 7, 16 Chen’s Lie algebra, 14 coexact forms, 16, 34 completion (inverse limit), 12, 14 current, 67 cycle, 78, 84
Jacobian manifold, 45 Jacobian variety, 45 Kahler manifold, 32 Kahler metric, 34
descending central series, 19 diagonal homomorphism, 9, 11 Dirac current, 67
Laplace operator, 30 lattice (discrete cocompact) subgroup, 26 linking number, 73, 76, 84-86, 89 lower central series, 7
Euclidean heat kernel, 76 Fermat quartic curve, 54 first Chern form, 69 free Lie algebra, 9
Maurer-Cartan I-form, 3 modular curve, 54
Green’s currents, 93 Green’s forms, 91 group homology, 22 group-like element, 11
orientations, 95 period matrix, 60 107
108
Iterated I n t e p l s and Cycles on Algebraic Manifolds
Poincare-Lelong formula, 69 primitive element (of Kahler manifold cohomology), 46 primitive elements (in a Hopf algebra), 11 quadratic differentials, 48 Quillen homomorphism Q, 8, 20, 21 Regularity theorem, 70 Schiffer variation, 47 spherical homology classes, 23 Torelli space, 47 universal enveloping algebra, 8-10
